ON A GENERALIZED, DOUBLY PARABOLIC

ON A GENERALIZED, DOUBLY PARABOLIC KELLER-SEGEL SYSTEM IN ONE SPATIAL DIMENSION ´ JAN BURCZAK AND RAFAEL GRANERO-BELINCHON Abstract. In this paper we study a Keller-Segel system with diffusion given by fractional laplacians in one spatial dimension. We obtain several local and global well-posedness results. In presence of a logistic term, this model is known for exhibit a spatio-temporal chaotic behaviour where a number of peaks emerge. We also study the dynamical properties of the system with the logistic term. In particular, we prove the existence of an attractor and provide a bound on the number of peaks that the solution may develop. These results generalize the known results where the diffusion is local. Finally, we perform a numerical analysis suggesting that there is a finite time blow up if the diffusion is weak enough even in the presence of a damping logistic term.

Contents 1. Introduction 1.1. Motivation 1.2. Plan of the paper and overview of our results 1.3. Novelties 2. Some prior results 2.1. Keller-Segel system with classical diffusion 2.2. Spatio-temporal chaos 2.3. Non-standard diffusions 3. Preliminaries 3.1. Singular integral operators and functional spaces 3.2. Sobolev embeddings and their constants 3.3. Notation 3.4. A notion of solution 4. Statement of results 4.1. Local-in-time existence, regularity and continuation criteria 4.2. Global-in-time solutions 4.3. Instantaneous analyticity 4.4. Large-time dynamics 4.5. Numerical study 5. Proof of Theorem 1: Local existence 6. Proof of Theorem 2: Continuation criterion 7. Proof of Theorem 3: Global existence using the Wiener’s algebra 8. Proof of Theorem 4: Global existence for α > 1 9. Proof of Theorem 5: Smoothing effect 10. Proof of Theorem 6: Existence of the attractor 11. Proof of Theorem 7: The number of relative maxima 1

2 2 4 5 5 5 6 6 7 7 7 8 8 8 8 9 10 11 13 13 15 17 19 27 32 34

´ J. BURCZAK AND R. GRANERO-BELINCHON

2

12. Numerical study 12.1. Algorithm 12.2. Results Appendix A. Auxiliary Lemmas Appendix B. Explicit expression for the constants Acknowledgments References

35 35 36 40 41 42 42

1. Introduction This paper is devoted to studies of the following generalized, doubly parabolic (τ = 1) Keller-Segel-type system with a logistic term (r ≥ 0) (1) (2)

∂t u = −µΛα u + ∂x (uΛβ−1 Hv) + ru(1 − u),

τ ∂t v = −νΛβ v − λv + u,

√ on T, the one dimensional periodic torus, where Λ = −∆ (for basic notation and definitions, see Section 3). In (1)-(2) we take parameters ν, µ, λ, β > 0, α, r ≥ 0 and nonnegative initial data u0 and v0 . We will refer to (1)-(2) with τ = 0 as to the parabolic-elliptic system and with τ = 1 as to the doubly parabolic one. A similar model has been mentioned by Biler & Wu, see [9, Section 5]. The system (1)-(2) with τ = 0 and α = 1, β = 2 is strictly connected to the one considered by us in [14], i.e. ∂t u = ∂x (−µ(u)Hu + u∂x v) + ru(1 − u), 0 = −∂x2 v + u − hui,

where we need to take a constant µ(u) ≡ µ. 1.1. Motivation. 1.1.1. Mathematical biology. Our interest in the system (1)-(2) stems from the mathematical studies of chemotaxis initiated by Keller & Segel in [32]. Chemotaxis is a chemically prompted motion of cells with density u towards increasing concentrations of a chemical substance with density v. For instance, in the case of the slime mold Dictyostelium Discoideum, the signal is produced by the cells themselves and cell populations might form aggregates in finite time. Chemotaxis also takes place in certain bacterial populations, such as of Escherichia coli and Salmonella typhimurium, and it results in their arrangement into a variety of spatial patterns. During embryogenesis, chemotaxis plays a role in angiogenesis, pigmentation patterning and neuronal development. It is also related to tumor growth. Specifically, in presence of the logistic term, this model is of particular importance in view of its relationship with the three-component urokinase plasminogen invasion model (see Hillen, Painter & Winkler [29]). Let us observe that the cell kinetics model M8 in Hillen & Painter [28], that describes a bacterial pattern formation or cell movement and growth

GENERALIZED KELLER-SEGEL

3

during angiogenesis, reads (3) (4)

∂t u = µ∆u − ∇(u∇v) + ru(1 − u) ∂t v = ν∆v − λv + u.

System (3)-(4) in one dimension is especially close to our system (1)-(2), since it is given by choosing α = β = 2 in (1)-(2) with τ = 1. The parabolic-elliptic (τ = 0) version of the system (3)-(4) is close to astrophysical models of a gravitational collapse. It is very similar in spirit to the Zel’dovich approximation [51] used in cosmology to study the formation of large-scale structure in the primordial universe, see also Ascasibar, Granero-Belinch´ on & Moreno [1]. It is also connected with the Chandrasekhar equation for the gravitational equilibrium of polytropic stars, statistical mechanics and the Debye system for electrolytes, see Biler & Nadzieja [8]. A more detailed presentation of some results on systems of type (3)-(4) and (1)-(2) follows in Section 2. 1.1.2. Fractional diffusion. The importance of the fractional diffusion generalization (1)-(2) of (3)-(4) is twofold. Primarily, there is a serious mathematical interest involved. To explain this point, let us recall that chemotaxis systems model two opposite phenomena: one is diffusion of cells due to their random movements, the other is their tropism toward higher concentrations of a chemical that may result in their aggregations. Hence it is mathematically interesting to establish the minimal strength of diffusion that overweights the chemotactic forces, hence giving, roughly speaking, the global existence of regular solutions or, equivalently, to study the maximal strength of diffusion that does not prevent blowup. Let us recall that for the parabolic-elliptic in two space dimensions the standard diffusion ∆ is critical; moreover the exact initial mass ku0 kL1 that divides the regimes of global existence and of blowup has been computed, compare for instance Bournavas & Calvez [12] and its references. Let us remark here that the blowup phenomenon together with the mass threshold was shown by Nagai [37]. For the doubly parabolic case in two space dimensions the situation is analogous, but here the available results are much later and less complete, see Mizoguchi [38] and references therein. Hence one may argue that the doubly parabolic case is substantially more difficult than the parabolic-elliptic one. For some more results, compare Section 2. In the one-dimensional case, the standard diffusion is strong enough to give the global existence; on the other hand, for d > 2 it is too weak. In this context it is mathematically interesting to find, for a fixed space dimension d, a ”critical” diffusive operator that sits on the borderline of the blowup and global-in-time regimes. There are at least two approaches to this problem, both justified from the point of view of applications. One is to consider the semilinear diffusion ∇ · (µ(u)∇u), see for instance Bedrossian, Rodriguez & Bertozzi [5], Blanchet, Carrillo & Lauren¸cot [10], Cie´slak & Stinner [18], Burczak, Cie´slak & Morales-Rodrigo [13], Cie´slak & Lauren¸cot [17] ad Tao & Winkler [42]. Another one is to replace the standard diffusion with the fractional one. In such a case there is a strong evidence that the


4

Peaks emerging

Peaks merging

5

18

t=0.19 t=0.59 t=0.79 t=0.99

4.5 4

t=1.19 t=1.59 t=1.79 t=1.99

16 14

3.5 12

u(x,t)

u(x,t)

3 2.5

10 8

2 6 1.5 4

1

2

0.5 0

−3

−2

−1

0

x

1

2

3

0

−3

−2

−1

0

1

2

3

x

Figure 1. Evolution in the case α = β = 1. √ half-laplacian Λ = −∆ is worth studying; for more on this, see subsection 2.3. We focus on the latter approach and one dimension. Let us mention here that the logistic term generally helps the global exison tence, see Tello & Winkler [43], Winkler [47], Burczak & Granero-Belinch´ [14]. However, in view of our interest in large-time behavior of solutions to (1)-(2), we include the logistic term in our considerations here mainly due to the context in which it appears in [41], namely the spatio-temporal chaos. Apart from the outlined mathematical interest in fractional diffusion systems, it is also believed that they can be useful for modelling the feeding strategies of microzooplancton, see Bournaveas & Calvez [11], Escudero [22] and the references therein. 1.2. Plan of the paper and overview of our results. In Section 2 we present some more known results on the Keller-Segel-type systems. Section 3 introduces basic notation, function spaces and our notion of solution. Next, we provide precise statements of our main results as well as some additional remarks in Section 4. The following sections contain proofs of our statements. In Section 5 we prove local existence of solutions to (1)-(2), while in Section 6 we show continuation criteria. Next, in Sections 7 and 8, we study the global-in-time existence. In particular, we prove global existence for the hypoviscous case α = 1, provided an explicit smallness condition for initial data holds and µ > 1. We also show global existence for arbitrary initial data in the case α > 1, β ≥ α/2. Furthermore, if 1 < α ≤ β ≤ 2, r, λ > 0, we obtain that the solutions remain bounded for every time. In Section 9 we study the smoothing properties of the systems (1)-(2), including an instantaneous gain of analyticity of the solution to (1)-(2). In Sections 10 and 11 we study existence of an attractor and the dynamical properties of (1)-(2) (for parameters α, β large enough). The solution in a neighborhood of this attractor develops a number of peaks that eventually merge themselves while other peaks emerge, see Figure 1. We are able to estimate analytically the number of these peaks. The aforementioned results are presented in Figure 2.


5

Bifurcation diagram for µ,ν, r,λ>0 2

Global well−posedness & Boundedness

1.8

1.6

Global well−posedness, Boundedness & Compact attractor

1.4

β

1.2

Global well−posedness

1

0.8

Local well−posedness 0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

α

1.2

1.4

1.6

1.8

2

Figure 2. Scheme assuming µ, ν, r, λ > 0. Finally, in Section 12, we perform a numerical study of (1)-(2) and provide numerical evidence of the finite time blow up in the case α = 0.5, β = 1. 1.3. Novelties. To the best of our knowledge, there are not many regularity results for the doubly parabolic fractional Keller-Segel system. Hence the generalization of the parabolic-elliptic global existence results of [11], [22], [14] to the system (1)-(2), even with β = 2 (the standard chemotactic term) and r = 0, appears to be new. In particular, we prove global existence and boundedness of classical solutions with no restriction on size of initial data in the subcritical regime α > 1 and with a restriction in the critical case α > 1. The restriction of the latter result is explicit and of the same order as the other parameters present in the system. In its proof we use the Wiener’s algebra approach, which seems to be new in the Keller-Segel context. Nevertheless we must admit that our smallness condition is quite stringent in the sense that it affects the entire Wiener’s algebra norm. The dynamical properties of the system are only known when α = β = 2, as far as we know. Moreover, the bound on the number of peaks seems new even in the classical α = β = 2 case. 2. Some prior results Let us now present some literature concerning the Keller-Segel-type systems, in addition to that mentioned in subsection 1.1. 2.1. Keller-Segel system with classical diffusion. There is a huge literature on the mathematical study of (3)-(4) and its parabolic-elliptic counterpart (τ = 0). Consequently, the list below is far from being exhaustive. The global existence of solutions to to (3)-(4) have been proved (under certain conditions) by many authors. In particular, Kozono & Sugiyama [35] showed the global existence and decay of solutions to (3)-(4) corresponding to small initial data in d = 3 and with 1 < r < 1.5 (see also [34]). Biler, Guerra & Karch [7] recently proved that for every finite Radon measure there exist τ0 and a global in time mild solution for (3)-(4) with τ > τ0 .

6


Corrias & Escobedo [21] proved that if the initial data (u0 , v0 ) is small in L1 (R2 ) × H˙ 1 (R2 ) there exists a global solution. This result was recently generalized by Cao [15]. Osaki & Yagi [40] and Osaki, Tsujikawa, Yagi & Mimura [39] obtained the existence of an exponential attractor while Hillen & Potapov [30], using different techniques, also showed the global existence of solutions. Winkler & Tello [43] proved the global existence of weak solutions for the parabolic-elliptic case with logistic term (τ = 0, r > 0) for arbitrary 0 ≤ u0 ∈ L∞ ; see also [44]. Winkler [47] showed that there exists a global in time solution for the doubly parabolic case with a sufficiently strong logistic parameter r. He also obtained global weak solutions and studied the regularizing properties starting from merely u0 ∈ L1 initial data in [46]. Some finite time singularities results for solutions corresponding to certain initial data can be found in [48, 49] by Winkler. 2.2. Spatio-temporal chaos. A remarkable feature of the model (1)-(2) is its spatio-temporal chaotic behaviour. In particular, the numerical solutions reported Painter & Hillen [41] for the system (3)-(4) develop a number of peaks that emerge and, eventually, mix with other peaks. These peaks are maxima of u, v that are very close to a region with their slope bigger than one. This phenomenon materializes in the numerical study of the system (1)-(2) with different values of α, β < 2 (see Figure 1 for the case α = β = 1 and Section 12.). As noted by Winkler in [47], the dynamical features of Keller-Segel models in high dimensions, in particular the existence of global attractors and bounded solutions, is an important topic. 2.3. Non-standard diffusions. The case with a nonlinear diffusion has been studied by several authors. See for instance Bedrossian & Rodriguez [4] Bedrossian, Rodriguez & Bertozzi [5], Blanchet, Carrillo & Lauren¸cot [10] and Burczak, Cie´slak & Morales-Rodrigo [13]. The case with fractional powers of the laplacian instead of local derivatives (α ∈ (0, 2) and β = 2) has been addressed by several authors. In particular, for the parabolic-elliptic case, Escudero [22] proved the boundedness of solutions in the one dimensional case with α > 1, while Li, Rodrigo & Zhang [36] proved finite time singularities by constructing a particular set of initial data showing this behaviour. These authors also proved that any L1t L∞ x bounded solution is global (see also [1]). Bournaveas & Calvez [11] studied the one-dimensional case with 0 < α < 1 and obtained the finite time blowup of solutions corresponding to big initial data and global solutions corresponding to small initial data. They also prove global existence for small data in the case α = 1, but here the problem of behavior of solutions emanating from large data remains open. Similar results were shown by Ascasibar, Granero-Belinch´ on & Moreno in [1]. The doubly parabolic case with fractional operators has been addressed by Biler & Wu [9] and Wu & Zheng [50]. In particular, these authors proved local existence of solutions, global existence of solutions for initial data satisfying some smallness requirements and ill-posedness in a variety of Besov spaces. For a nonlinear, fractional diffusion see also Granero-Belinch´ on & on [14]. Orive [26] and Burczak & Granero-Belinch´


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3. Preliminaries Here we gather some basic terms used in what follows. We define Z 1 hf i = f (x)dx, T = [−π, π]. |T| T 3.1. Singular integral operators√and functional spaces. We write H for the Hilbert transform and Λ = −∆, i.e. s u(k) = |k|s u d d Hu(k) = −i sgn(k) uˆ(k), and Λ ˆ(k),

where ˆ· denotes the usual Fourier transform. Notice that Λ = ∂x H in one √ 2 d dimension and Hu(0) = 0. The differential operator Λs = ( −∂x )s is defined by the action of the following kernels (see [19] and the references therein): Z X Z f (x) − f (y) f (x) − f (y) s dy + cs dy, (5) Λ f (x) = cs p.v. 1+s 1+s T |x − y + 2kπ| T |x − y| k∈Z\{0}

where cs > 0 is a normalization constant. In particular, in one dimension for s = 1 Z 1 f (x) − f (y) Λf (x) = p.v. dy. 2 2π T sin ((x − y)/2) Remark 1. Notice that given v ∈ L2 (T), since hHvi = 0, Λβ−1 Hv ∈ L2 (T)

even if β < 1. We write H s for the usual L2 -based Sobolev spaces with the norm kf k2H s = kf k2L2 + kf k2H˙ s ,

kf kH˙ s = kΛs f kL2 .

The Wiener’s algebra is defined as (6)

A(T) = {periodic functions f such that fˆ ∈ l1 },

i.e., is the set of functions with absolutely convergence Fourier series. For a periodic function u, we define the Wiener’s algebra-based seminorms: X |u|s = |k|s |ˆ u(k)|. k∈Z

3.2. Sobolev embeddings and their constants. Along the paper we are going to use different forms of Sobolev embedding (all of them classical). For the sake of clarity, we collect here these inequalities (and denote their constants) that are more often used. Assuming α > 1, we have (7)

1 (α)kΛ1−α/2 f kL2 , kf kL2/(α−1) ≤ CSE 2 kf kL∞ ≤ CSE (α)kΛα/2 f kL2 ,

(8) (9) (10)

kf k

L

2+ 2α−2 2−α

3 ≤ CSE (α)kΛ(α−1)/2 f kL2 ,

4 (α)kf kH˙ α . kf kW α/2,∞ ≤ CSE


8

3.3. Notation. We write Tmax for the maximum lifespan of the solution. For a given initial data (u0 , v0 ), we define N = max{ku0 kL1 (T) , 2π}. 3.4. A notion of solution. Let u0 (x), v0 (x) ≥ 0 be the initial data for the system (1)-(2). Then we define its solution as follows Definition 1. Let 0 < T < ∞ be a positive parameter. The couple (u, v) ∈ L∞ ([0, T ], L2 (T)) × L∞ ([0, T ], H β/2 (T))

is a solution of (1)-(2) if Z Z TZ α β−1 [∂t φ−µΛ φ]u+∂x φ(uΛ Hv)+φru(1−u)dxdt− φ(x, 0)u0 dx = 0, 0

T

T

Z

T 0

Z

T

[∂t ϕ − νΛβ ϕ − λϕ]v + ϕudxdt −

for all test functions φ, ϕ ∈ Cc∞ ([0, T ) × T).

Z

ϕ(x, 0)v0 dx = 0,

T

Definition 2. If a solution (u, v) verifies the previous definition for every 0 < T < ∞, this solution is called a global solution. Observe, that our notion of a global solution does not involve T = ∞. In particular, our global solution may a priori become arbitrarily large as time tends to infinity. 4. Statement of results 4.1. Local-in-time existence, regularity and continuation criteria. First, we have the following result Theorem 1. Let s ≥ 3, ν, λ > 0 and 0 < β, α ≤ 2. If (u0 , v0 ) ∈ H s (T) × H s+β/2 (T) is the initial data for equation (1)-(2), then it admits a unique solution 0 ≤ u ∈ C([0, Tmax (u0 , v0 )], H s (T)) ∩ L2 ([0, Tmax (u0 , v0 )], H s+α/2 (T)), 0 ≤ v ∈ C([0, Tmax (u0 , v0 )], H s+β/2 (T)) ∩ L2 ([0, Tmax (u0 , v0 )], H s+β (T)). Next, we prove the following continuation criteria, slightly stronger than the condition in [49, Lemma 2.1], Theorem 2. Assume that, for a finite time T and initial data (u0 , v0 ) ∈ H s (T) × H s+β/2 (T), s ≥ 3, the solution to (1)-(2) satisfies Z T kΛβ v(s)kL∞ + k∂x u(s)kL∞ ds < ∞, 0

then this solution can be continued up to time T +δ for a small enough δ > 0. 1 , α ≥ β, then, the previous condition can be replaced by Moreover, if µ ≥ 2ν Z T ku(s)k2L∞ + kΛβ v(s)kL∞ + ku(s)kL∞ kΛβ v(s)kL∞ ds < ∞. 0


9

Hence if (u, v) is a solution showing finite time existence with being Tmax its maximum lifespan, then we have lim sup kΛβ v(t)kL∞ + k∂x u(t)kL∞ = ∞. t→Tmax

And, if µ ≥

1 2ν ,

α ≥ β, the previous equation is replaced by lim sup ku(t)kL∞ + kΛβ v(t)kL∞ = ∞. t→Tmax

Let us emphasize that the above results do not involve any extra assumptions on the values of parameters α, β, r, λ. They should be compared with the results in [47, Lemma 1.1]. 4.2. Global-in-time solutions. 4.2.1. Small data result for α ≥ 1. Using a Wiener’s algebra approach we obtain a global solution for small, periodic initial data. Recall that the Wiener’s algebra is defined as in (6). Theorem 3. Let (u0 , v0 ) ∈ H 3 (T) × H 3+β/2 (T) and assume 1 ≤ β ≤ 2 ≤ 1 + α as well as r = 0, µ > 1 in the system (1)-(2). Then, if the initial data satisfy |u0 |1 + |v0 |β < min{µ − 1, ν − hu0 i, λ/2}, the solution is global and |u(t)|1 + |v(t)|β ≤ |u0 |1 + |v0 |β . This result has the same flavour as [2, 35]. The case α = 1 is particularly interesting because for the case α > 1 we prove below the existence of global solutions corresponding to arbitrary large initial data. Notice that the constant in the smallness condition depends explicitly on the parameters present in the problem and ku0 kL1 . Theorem 3 is stated for the case r = 0. Let us recall, that in [14] there is a result for a similar, parabolic-elliptic system that presents an interplay between the admissible size of the initial data and the logistic parameter r. 4.2.2. Results for α > 1. For 1 < α, α/2 ≤ β ≤ 2 we have Theorem 4. Let µ, ν > 0, 2 ≥ α > 1, r, λ ≥ 0, 2 ≥ β ≥ α/2 and the initial data (u0 , v0 ) ∈ L2 (T) × H β−α/2 (T) be given. Then there exists at least one global in time weak solution corresponding to (u0 , v0 ) satisfying u ∈ L∞ ([0, T ], L2 (T)) ∩ L2 ([0, T ], H α/2 (T))

∀ T < ∞,

v ∈ L∞ ([0, T ], H β−α/2 (T)) ∩ L2 ([0, T ], H 3β/2−α/2 (T)) ∀ T < ∞. If, in addition, the initial data (u0 , v0 ) ∈ H kα (T) × H kα+β/2 (T), k ∈ N, kα ≥ 3, then there exists a unique global in time solution corresponding to (u0 , v0 ) that enjoys u ∈ C([0, T ], H kα (T))

∀ T < ∞,

v ∈ C([0, T ], H kα+β/2 (T)) ∀ T < ∞. Moreover, in the case r > 0, α ≤ β, there exist positive numbers T ∗ , S(·) such that ku(t + 1)k2H˙ kα/2 ≤ S(H˙ kα/2 ) ∀ t ≥ T ∗ , 0 ≤ kα.


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From Theorem 4 follows in particular that lim sup ku(t)k2H α/2 ≤ S(H α/2 ) t→∞

S(H α/2 )

with given by (42). Furthermore it implies that there exists C, depending on the parameters present in the problem and on the initial data, such that max {ku(t)k2H kα/2 + kv(t)k2H β+(k−1)α/2 } ≤ C, 0≤t 0, r ≥ 0 and the initial data (u0 , v0 ) ∈ H 3 (T)×H 4 (T), the solution (u, v) of (1)-(2) becomes real analytic for every 0 < t < T˜. Furthermore, the complex extension of (u, v) becomes complex analytic in the growing in time, complex strip Sω with ω ≤ ω0 and we have the bounds √ √ ku(t)kL∞ (Sω ) ≤ 2ku0 kL∞ (T) , kv(t)kL∞ (Sω ) ≤ 2kv0 kL∞ (T) . Remark 3. If in addition min{α, β} > 1, the restriction ω ≤ ω0 can be relaxed. Theorem 5 is local in time. However, using a classical bootstrapping argument, the analyticity of (u, v) in a complex strip around the real axis (possible with a very small width of the strip) can be obtained for every positive time t > 0. More precisely, we have Corollary 1. If α, β > 1, and min{µ, ν} > 0, r ≥ 0, the solutions (u, v) ∈ H 3 (T) × H 4 (T) to the problem (1)-(2) are real analytic for every 0 < t. Moreover, it holds Corollary 2. If α, β ≥ 1, and min{µ, ν} < 0, the problem is ill-posed, i.e. there are solutions (u, v) to the problem (1)-(2) such that ku0 kH 3 (T) + kv0 kH 4 (T) < ǫ and lim sup ku(t)kH 3 (T) + kv(t)kH 4 (T) = ∞, t→δ−

for every ǫ > 0 and small enough δ > 0. 4.4. Large-time dynamics. We are interested in the existence of attractors for (1)-(2) and their dynamical properties. 4.4.1. Existence of attractor. We answered the question about the existence of a connected, compact attractor in the following theorem. Theorem 6. Given r > 0, 2 ≥ β ≥ α ≥ 8/7, the system (1)-(2) has a maximal, connected, compact attractor in the space H 3α (T) × H 3α+β/2 (T).


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4.4.2. Estimates of number of peaks. We can apply Theorem 5 to study certain dynamical properties of the system (1)-(2). In particular, applying Theorem 5 together with the complex analyticity of the solution (u, v), we can obtain a bound on the number of peaks. Theorem 7. Let N ≥ 3, N ∈ N, 2 ≥ α, β ≥ 1, µ, ν > 0, λ, r ≥ 0 and the initial data (u0 , v0 ) ∈ H 3 (T) × H 4 (T) be given and write W=

ω0 T˜ , N

where ω0 and T˜ are defined in (11) and (12) respectively. Then, for any ǫ > 0, 0 < T˜/(N − 1) < t < T˜, T = Iǫu ∪ Rǫu = Iǫv ∪ Rǫv ,

where Iǫu , Iǫv are the union of at most [ 4π W ] intervals open in T, and u • |∂x u(x)| ≤ ǫ, for all x ∈ Iǫ , √ 2(N −1)ku0 kL∞ (T) 2 2π u • #{x ∈ Rǫ : ∂x u(x) = 0} ≤ log 2 W log , Wǫ • |∂x v(x)| ≤ ǫ, for all x ∈ Iǫu , • #{x ∈

Rǫv

: ∂x v(x) = 0} ≤

2 2π log 2 W

log

√

2(N −1)kv0 kL∞ (T) Wǫ

.

Notice that Theorem 7 gives us an estimate of the number of peaks appearing in the evolution (and reported in the numerical simulations). Indeed, we have the following corollary. Corollary 3. Let r > 0, 2 ≥ β ≥ α ≥ 8/7 and (u, v) be a solution in the attractor, then the number of peaks for u can be bounded as √ 12πK1 2 α/2 log 6 2K1 CSE (α)S(H ) , #{peaks for u} ≤ log 2 where S(H α/2 ) and K1 are defined in (42) and (43), respectively.

In [41], the authors perform a numerical study of the case α = β = 2, µ = ν, r = λ and different values of λ and ν. They take initial data satisfying ku0 kL∞ = 1, kv0 kL∞ ≤ 1.01. We can use our previous results to give an analytical bound on the number of peaks that the solutions in [41] develop: Corollary 4. Let (u, v) be the solution corresponding to the initial data in [41], then the number of peaks for u and v can be bounded as follows √ 12πK1 log 6 2K1 , #{peaks for u} ≤ log 2 √ 12πK1 log 6 2K1 1.01 , #{peaks for v} ≤ log 2 where K1 = K1 (2, 2, ν, ν, λ, λ) is defined in (43). Let us finally emphasize that presence of peaks is in no connection with a possibility of a blow up of solution.


13

4.5. Numerical study. In the numerical part (Section 12), we present first our simulations of emerging and merging peaks. The main conclusion from our numerical study for further analysis is that, even in presence of a damping logistic term, our system may develop finite time singularities for certain parameters. In particular, our numerics suggest that for α = 0.5, β = 1 and a sufficiently strong nonlinear term (measured by chemical sensitivity χ there, compare the system (34) - (35)), the solution blows up in a finite time. Furthermore, our numerical solutions agree with the continuation criterion in Theorem 2 in the sense that the spatial norms of the numerical solutions k∂x ukL∞ are not integrable. 5. Proof of Theorem 1: Local existence We prove now our local well-posedness result. Proof of Theorem 1. We prove the case s = 3, the other cases being similar. Part 1. (a priori estimates) We compute 1 d kvk2L2 ≤ −νkvk2H˙ β/2 − λkvk2L2 + kukL2 kvkL2 , 2 dt Z 1 d β/2 3 2 Λβ ∂x3 v∂x3 ∂t v kΛ ∂x vkL2 = 2 dt T = −νkvk2H˙ 3+β − λkvk2H˙ 3+β/2 + kukH˙ 3 kvkH˙ 3+β 2 ν ≤ − kvk2H˙ 3+β − λkvk2H˙ 3+β/2 + kuk2H˙ 3 , 2 ν Z Z 1d kuk2L2 = −µkuk2H˙ α/2 − ∂x uuΛβ−1 Hvdx + r u2 (1 − u)dx 2 dt T T Z Z 1 2 2 2 β = −µkukH˙ α/2 + u Λ vdx + r u (1 − u)dx 2 T T ≤ ku(t)k2L2 kΛβ vkL∞ + rkuk2L2 ,

and

Z 1 d 3 2 2 k∂ uk 2 = −µkukH˙ 3+α/2 + ∂x3 u∂x4 (uΛβ−1 Hv)dx 2 dt x L T Z + r ∂x3 u∂x3 (u(1 − u))dx. T

The higher order terms are Z ∂x3 u∂x4 uΛβ−1 Hvdx ≤ kΛβ vkL∞ kuk2H˙ 3 , J1 = ZT ∂x3 uuΛβ ∂x3 vdx ≤ kΛβ ∂x3 vkL2 kukH˙ 3 kukL∞ , J2 = T Z J3 = r (∂x3 u)2 (1 − 2u)dx ≤ rk∂x3 u(t)k2L2 . T

Using

Z

T

4

(∂x f ) dx = −3

Z

∂x2 f (∂x f )2 f,


14

together with Hölder’s inequality, we obtain the following Gagliardo-Niremberg inequality k∂x f k2L4 ≤ 3kf kL∞ k∂x2 f kL2 .

(15)

Due to (15), the lower order terms can be bounded as Z ∂x3 u∂x3 uΛβ vdx ≤ kΛβ vkL∞ kuk2H˙ 3 , J4 = T

J5 =

Z

T

∂x3 u∂x2 uΛβ ∂x vdx ≤ kΛβ ∂x vkL4 kukH˙ 3 k∂x2 ukL4

1.5 0.5 β 2 0.5 ≤ CkΛβ vk0.5 L∞ kΛ ∂x vkL2 kukH˙ 3 k∂x ukL∞ , Z ∂x3 u∂x uΛβ ∂x2 vdx ≤ kΛβ ∂x2 vkL2 kukH˙ 3 k∂x ukL∞ , J6 = T

and J7 = 6r

Z

T

0.5 0.5 ∂x3 u∂x u∂x2 udx ≤ 6rkukH˙ 3 k∂x ukL4 k∂x2 ukL4 ≤ Ckuk2H 3 kukL ∞ k∂x ukL∞ .

We define the energy We have

E = kuk2H 3 + kvk2H 3+β/2 .

d E ≤ C(E + 1)3 + kΛβ ∂x3 vkL2 kukH˙ 3 kukL∞ dt 2 ν − kvk2H˙ 3+β − λkvk2H˙ 3+β/2 + kuk2H˙ 3 − µkuk2H˙ 3+α/2 2 ν ν 4 2 (16) ≤ C(ν)(E + 1) − µkukH˙ 3+α/2 − kvk2H˙ 3+β . 4 Due to the previous inequality (16), we obtain the desired bound for the 3+α/2 × L2t Hx3+β . energy. Moreover, from (16), we get that (u, v) ∈ L2t Hx Part 2. (existence) Once we have the energy estimates, we consider a family of Friedrichs mollifiers Jǫ and define the regularized initial data uǫ (x, 0) = Jǫ ∗ u0 (x) ≥ 0, v ǫ (x, 0) = Jǫ ∗ v0 (x) ≥ 0,

and the regularized problems

∂t uǫ = −µJǫ ∗ Λα Jǫ ∗ uǫ + Jǫ ∗ ∂x · (Jǫ ∗ uǫ (Λβ−1 HJǫ ∗ v ǫ )) + ruǫ (1 − uǫ ) ∂t v ǫ = −Jǫ ∗ νΛβ Jǫ ∗ v ǫ − λJǫ ∗ v ǫ + Jǫ ∗ uǫ .

Applying Picard’s Theorem in H 3 × H 3+β/2 , we find a solution (uǫ , v ǫ ) to these approximate problems. These solutions exists up to time T ǫ . Furthermore as (uǫ , v ǫ ) verify the same energy estimates, we can take T = T (u0 , v0 ) independent of the regularizing parameter ǫ. This concludes the existence part. Part 3. (uniqueness) To prove the uniqueness we argue by contradiction. Let us assume that there are two different solutions corresponding to the same initial data (u0 , v0 ) ∈ L2 × H β/2 . We write (ui , vi ), i = 1, 2 for these solutions and define u ¯ = u1 − u2 , v¯ = v1 − v2 . Then we have the bounds d u(t)k2L2 , v (t)k2H˙ β ≤ c(ν)k¯ k¯ v (t)k2H β/2 + νk¯ dt


15

Z d 2 β−1 β−1 k¯ u(t)kL2 ≤ 2 ∂x u ¯[¯ uΛ Hv1 + u2 Λ H v¯]dx dt T

u(t)kL2 k∂x u2 (t)kL∞ kΛβ−1 v¯(t)kL2 ≤ k¯ u(t)k2L2 kΛβ v1 (t)kL∞ + k¯ + k¯ u(t)kL2 ku2 (t)kL∞ kΛβ v¯(t)kL2 .

We use β − 1 ≤ β/2 and Young’s inequality to get

d v (t)k2H β/2 ) u(t)k2L2 + k¯ v (t)k2H β/2 ) ≤ (k¯ (k¯ u(t)k2L2 + k¯ dt × c(ν) + kΛβ v1 (t)kL∞

1 + k∂x u2 (t)kL∞ + c(ν)ku2 (t)k2L∞ 2

Finally we get

.

v0 k2H β/2 ) u0 k2L2 + k¯ v (t)k2H β/2 ≤ (k¯ k¯ u(t)k2L2 + k¯ × e[c(ν)t+

Rt 0

kΛβ v1 (s)kL∞ +0.5k∂x u2 (s)kL∞ +c(ν)ku2 (s)k2L∞ ]dt

From the last inequality we obtain the uniqueness. Part 4. (preservation of sign) To finish the proof of the entire Theorem 1, we need to show that for the non-negative initial data the solution remains non-negative as well. To obtain pointwise bounds we apply the techniques developed in [14, 19, 20, 26, 24] and the references therein. Let u(x, t) be a classical solution with a non-negative initial data and write xt ∈ T for the point such that minx u(x, t) = u(xt , t). Evaluating the equation (1) at the point of minimum and using the kernel expression for Λα , we have h i d u(xt , t) ≥ u(xt , t) Λβ v(xt , t) + r(1 − u(xt , t)) , t ≥ 0, dt hence Rt β u(xt , t) ≥ u0 (x0 )e 0 Λ v(xy ,y)+r(1−u(xy ,y))dy . Thus, u(xt , t) ≥ 0 since u0 (x) ≥ 0 and we conclude the claim. For the equation (2) we can proceed similarly and we get Z t −λt −λt v(xt , t) ≥ v0 (x)e +e u(xs , s)ds ≥ 0. 0

Let us remark here that in the above proof and in the remainder of this text, we reinterpret certain terms in language of duality pairs, where there is no sufficient regularity to perform some intermediate computations. This applies in particular to the time derivative of a single (spatial) Fourier mode. 6. Proof of Theorem 2: Continuation criterion In this section, we proceed with the proof of Theorem 2: Proof of Theorem 2. Part 1. Let us write Z T kΛβ v(s)kL∞ + k∂x u(s)kL∞ ds = M < ∞. (17) 0

.


16

Now, assuming the finiteness of M , we need to obtain a bound for the energy E(t) = ku(t)k2H 3 + kv(t)k2H 3+β/2 . First notice that 1 d 1 kuk2L2 ≤ kΛβ vkL∞ kuk2L2 + rkuk2L2 , 2 dt 2 1d 1 kvk2L2 ≤ kuk2L2 . 2 dt 2 Thus sup ku(t)k2L2 ≤ ku0 k2L2 eM +rT ,

0≤t≤T

sup kv(t)k2L2 ≤ (ku0 k2L2 eM + kv0 k2L2 )eT .

0≤t≤T

Let xut denote the point where u(t) reaches its maximum and let xvt denote the point where v(t) reaches its maximum. Then, u(xut , t) and v(xvt , t) are Lipschitz functions and, as a consequence, applying Rademacher Theorem, are almost everywhere differentiable. Moreover, using the expressions for the kernels Λs together with the positivity of u and v, we have d u(xut , t) ≤ u(xut , t)Λβ v(xut , t) + ru(xut , t), dt d v(xvt , t) ≤ u(xvt , t). dt As a consequence, sup ku(t)kL∞ ≤ ku0 kL∞ eM +rT ,

0≤t≤T

sup kv(t)kL∞ ≤ kv0 kL∞ + ku0 kL∞ eM T.

0≤t≤T

Notice that to bound the lower order norms we have used merely Z T kΛβ v(s)kL∞ ds < ∞. 0

For the higher seminorm,

y(t) = ku(t)k2H˙ 3 + kv(t)k2H˙ 3+β/2 , due to energy estimates, we have dy ≤ c(M, ν)(kΛβ vkL∞ + k∂x ukL∞ + ku0 kL∞ eM +rT )y(t), dt and using Gronwall’s inequality we conclude the result in the case when M of (17) is finite. Part 2. To simplify notation we write Z T ˜. ku(s)k2L∞ + kΛβ v(s)kL∞ + ku(s)kL∞ kΛβ v(s)kL∞ ds = M 0

˜, The idea for this second part is similar. Assuming the boundedness of M it suffices to obtain global bounds for M defined in (17). To this end, we


17

˜ < ∞ to bound ku(t)kH 2 . Then we can use Sobolev’s are going to use M embedding to obtain the estimate for M . First we compute 1 d β 2 2 kΛ ∂x vkL2 +λkΛβ ∂x2 vk2L2 ≤ −νkΛ1.5β ∂x2 vk2L2 +kΛβ/2 ∂x2 ukL2 kΛ1.5β ∂x2 vkL2 , 2 dt thus 1d β 2 2 1 kΛ ∂x vkL2 + λkΛβ ∂x2 vk2L2 ≤ kΛβ/2 ∂x2 uk2L2 . 2 dt 2ν Now we have 1 d 2 2 k∂ uk 2 ≤ −µkΛα/2 ∂x2 uk2L2 + ckΛβ vkL∞ k∂x2 uk2L2 2 dt x L β 2 0.5 β 0.5 2 0.5 + ck∂x2 ukL2 kuk0.5 L∞ k∂x ukL2 kΛ vkL∞ kΛ ∂x vkL2

+ kukL∞ kΛβ ∂x2 vkL2 k∂x2 ukL2 + rk∂x2 uk2L2 + 2rk∂x2 ukL2 k∂x uk2L4 .

Using the previous bound and Young’s inequality, we get 1 1 d 2 2 ≤ −µkΛα/2 ∂x2 uk2L2 + kΛβ/2 ∂x2 uk2L2 k∂x ukL2 + kΛβ ∂x2 vk2L2 2 dt 2ν 1 kuk2L∞ k∂x2 uk2L2 +ckΛβ vkL∞ k∂x2 uk2L2 + 2λ β 0.5 +ck∂x2 uk2L2 kuk0.5 L∞ kΛ vkL∞ +c(λ)k∂x2 uk2L2 kukL∞ kΛβ vkL∞

+crk∂x2 uk2L2 (kukL∞ + 1) .

Hence we obtain a bound for the H 2 seminorm. In the same way we get a bound for the L2 norm. Since we have a bound for H 2 , using Sobolev embedding, we arrive at Z T Z T ˜ . ku(s)kH 2 ds ≤ cT ku0 kH 2 exp c(λ)M k∂x u(s)kL∞ ds ≤ c 0

0

7. Proof of Theorem 3: Global existence using the Wiener’s algebra Our aim here is to prove Theorem 3. We start this section with two preliminary results concerning lower order norms. The behaviour is quite different depending on the value of r. If r = 0, we have Lemma 1. Let (u0 , v0 ) be two non-negative, smooth initial data for equation (1)-(2) with r = 0. Then, the solutions (u, v) are non-negative functions. Moreover, if, in addition, the initial data (u0 , v0 ) are in L1 (T), the solutions (u, v) verify • ku(t)kL1 (T) = ku0 kL1 (T) ∀ 0 ≤ t ≤ Tmax ku0 kL1 (T) ku0 kL1 (T) + kv0 kL1 (T) − e−λt ∀ 0 ≤ t ≤ • kv(t)kL1 (T) = λ λ Tmax .

For the sake of brevity we do not write the proof. For r > 0 the analogous result reads (see also [29]).


18

Lemma 2. Let (u0 , v0 ) be two non-negative, smooth initial data for equation (1)-(2) with r > 0. Let us define N = max{ku0 kL1 (T) , 2π}. Then the solutions (u, v) verify • ku(t)kL1 (T) ≤ N , ∀ 0 ≤ t ≤ Tmax Rt 2 ∀ 0 ≤ t ≤ Tmax , 0 ku(s)kL2 (T) ds ≤ N t + 2N , • kv(t)kL1 (T) ≤ max{kv0 kL1 , N /λ}, ∀ 0 ≤ t ≤ Tmax . Proof. We take r = 1 without losing generality. The ODE for ku(t)kL1 is

d ku(t)kL1 = ku(t)kL1 − ku(t)k2L2 . dt Recalling Jensen’s inequality

(18)

ku(t)k2L1 ≤ 2πku(t)k2L2 , we get

d 1 ku(t)kL1 ≤ ku(t)kL1 1 − ku(t)kL1 . dt 2π From this inequality we conclude the first part of the result. Given t > 0, we integrate (18) between 0 and t and obtain Z t Z t ku(s)k2L2 ds, ku(s)kL1 ds − ku(t)kL1 − ku0 kL1 = 0

0

thus Z

0

t

ku(s)k2L2 (T) ds ≤ ku0 kL1 − ku(t)kL1 + sup ku(s)kL1 t ≤ N t + 2N , 0≤s≤t

and we conclude the second part. The bound for the L1 norm of v is straightforward and we get N N kv(t)kL1 (T) ≤ + kv0 kL1 (T) − e−λt , ∀ t ≥ 0, (λ > 0), λ λ or kv(t)kL1 (T) ≤ N t + kv0 kL1 (T) , ∀ t ≥ 0, (λ = 0).

Now we turn to the proof of Theorem 3. Recall that we assume there 1 ≤ β ≤ 2 ≤ 1 + α and µ > 1, r = 0. Proof of Theorem 3. We denote by fˆ(k) the k−th Fourier mode of the funcˆ(0, t) = hu0 i. We will study tion f . Then, as stated in Lemma 1 we have u the evolution of E(t) = |u(t)|1 + |v(t)|β .

Our goal is to obtain (under appropriate assumptions) the maximum principle (19)

E(t) ≤ E(0).


19

Having this together with Fourier series that imply X ∂x u = ij u ˆ(j)eijx ⇒ k∂x ukL∞ ≤ |u|1 , j

Λβ v =

X j

we arrive at

Z

T 0

|j|β vˆ(j)eijx ⇒ kΛβ vkL∞ ≤ |v|β ,

k∂x u(s)kL∞ + kΛβ v(s)kL∞ ds ≤ E(0)T.

Using the continuation argument in Theorem 2, we conclude the proof. It remains to obtain the maximum principle (19). The system (1)-(2) reads ¯ˆ(k) X k−j |k|u d ju ˆ(j) |ˆ u(k)||k| = −µ|k|1+α |ˆ u(k)| + |k − j|β−1 vˆ(k − j) dt |ˆ u(k)| |k − j| j

¯ |k|u ˆ(k) X + u ˆ(k − j)|j|β vˆ(j), |ˆ u(k)| j

v¯ˆ(k) β d |ˆ v (k)||k|β = −ν|k|2β |ˆ v (k)| − λ|ˆ v (k)||k|β + |k| u ˆ(k), dt |ˆ v (k)| so, using Fubini-Tonelli’s Theorem d |u(t)|1 ≤ −µ|u|1+α + 2|u|1 |v|β + |u|2 |v|β−1 + |u|0 |v|β+1 , dt and d E ≤ −µ|u|1+α + 2|u|1 |v|β + |u|2 |v|β−1 + |u|0 |v|β+1 − ν|v|2β − λ|v|β + |u|β . dt Using Young’s inequality and the assumptions we get d E ≤ (|v|β + 1 − µ)|u|2 + (|u|1 + hu0 i − ν)|v|2β + (2|u|1 − λ)|v|β dt ≤ (E + 1 − µ)|u|2 + (E + hu0 i − ν)|v|2β + (2E − λ)|v|β ,

thus, if

E(0) < min{µ − 1, ν − hu0 i, λ/2}, we obtain a decay (consequently, a global bound) for E(t).

8. Proof of Theorem 4: Global existence for α > 1 Now we proceed with the proof of the global existence of solutions for large data. Proof of Theorem 4. As α > 1, we can define α−1 = δ > 0 as a fixed 2 parameter. Recall that T is an arbitrary fixed number such that 0 < T < ∞. We will consider times 0 ≤ t ≤ T . Let us outline the proof: in the first three steps, we obtain a priori estimates. i.e. we assume there that we have a solution (u, v) as smooth as required. In Step 4, we construct the solutions as the limit of approximate problems satisfying the same a priori estimates as in Step 1,2 and 3. Finally, in Step 5, we obtain the existence of an absorbing set and, as a consequence, the boundedness of the solutions.


20

Step 1. showing

(a priori estimates I)

In this step we obtain estimates

u ∈ L∞ (0, T ; L2 (T)) ∩ L2 (0, T ; H α/2 (T))

v ∈ L∞ (0, T ; H β−α/2 (T)) ∩ L2 (0, T ; H 3β/2−α/2 (T)).

We compute the evolution of the L2 norm of u in the case r > 0. If r = 0 the proof is analogous. We get Z

Z 1 = −µ |Λ u| dx + u2 Λβ vdx + rku(t)k2L2 − rku(t)k3L3 2 T T 1 α/2 2 ≤ −µku(t)kH˙ α/2 + kΛ (u(t))2 kL2 kΛβ−α/2 v(t)kL2 2 2 +rku(t)kL2 − rku(t)k3L3

1 d ku(t)k2L2 2 dt

α/2

2

≤ −µku(t)k2H˙ α/2 + rku(t)k2L2 − rku(t)k3L3 +CKP (α)kukL∞ kukH˙ α/2 kΛβ−α/2 vkL2

≤ −µku(t)k2H˙ α/2 + rku(t)k2L2 − rku(t)k3L3

+CKP (α)hu(t)iku(t)kH˙ α/2 kΛβ−α/2 v(t)kL2 δ/(1+δ)

+CKP (α)CGN (α)ku(t) − hu(t)ikL1

×kΛβ−α/2 v(t)kL2 ,

(2+δ)/(1+δ)

ku(t)kH˙ α/2

where we have used Lemma 4 together with the following interpolation inequality (20)

δ/(1+δ)

|kukL∞ − hui| ≤ ku − huikL∞ ≤ CGN (α)ku − huikL1

1/(1+δ)

kukH˙ α/2 .

Using Young’s inequality and Lemmas 1 and 2, we obtain d ku(t)k2L2 dt

≤ −µku(t)k2H˙ α/2 + rku(t)k2L2 − rku(t)k3L3 + +

(21)

+

(CKP (α)CGN (α))

2+2δ δ

µ ku(t)k2H˙ α/2 2 2+2δ

2+2δ

(2N ) 1+δ kΛβ−α/2 v(t)kL2δ µ

(CKP (α)N )2 β−α/2 kΛ v(t)k2L2 . µ

Now, fix t > 0 and consider the equation for the k-th Fourier coefficient of v d vˆ(k, t) = −ν|k|β vˆ(k, t) − λˆ v (k, t) + u ˆ(k, t). dt Solving this ODE, we get (22)

(ν|k|β +λ)t

e

vˆ(k, t) = vˆ0 (k) +

Z

t 0

e(ν|k|

β +λ)s

u ˆ(k, s)ds.


21

As v0 ∈ H γ with γ = β − α/2 < β − 0.5, using (22), we get |k|β−α/2 e(ν|k|

β +λ)t

|ˆ v (k, t)| ≤ |k|β−α/2 |vˆ0 (k)| Z t β |k|β−α/2 e(ν|k| +λ)s |ˆ u(k, s)|ds + 0

≤ |k|β−α/2 |vˆ0 (k)| +

(23) hence Z

t

0

β |k|β−α/2 N e(ν|k| +λ)t , β ν|k| + λ

kΛβ−α/2 v(s)kpL2 ds ≤ tC(α, β, λ, ku0 kL1 (T) , kv0 kH β−α/2 (T) , ν, p).

Consequently, using Lemma 2, we have that Z µ t 2 ku(t)kL2 + ku(s)k2H˙ α/2 + rku(s)k3L3 ds 2 0 ≤ ku0 k2L2 + N t + 2N + tC(α, β, ν, λ, ku0 kL1 (T) , kv0 kH β−α/2 (T) ). In the case λ > 0 we obtain simply Z t Z t 2 2 2 ku(s)k2L2 ds. kv(s)kH˙ β/2 ds ≤ kv0 kL2 + c(λ) kv(t)kL2 + 2ν 0

0

Testing the equation for v against get

Λ2β−α v

and using self-adjointness we

1 d kv(t)k2H˙ β−α/2 +λkv(t)k2H˙ β−α/2 +νkv(t)k2H˙ 3β/2−α/2 ≤ kukH˙ α/2 kv(t)kH˙ 2β−α−α/2 . 2 dt As β ≤ 2 < 2α, we get 2β − α − α/2 ≤ 1.5β − α/2 and we can use Young’s and Poincaré’s inequalities to conclude this step. Step 2. (a priori estimates II) In this part we obtain estimates showing u ∈ L∞ (0, T ; H α/2 (T)) ∩ L2 (0, T ; H α (T)),

v ∈ L∞ (0, T ; H β/2+α/2 (T)) ∩ L2 (0, T ; H β+α/2 (T)).

Testing the equation for v against Λα+β v, we get 1 d kv(t)k2H˙ β/2+α/2 + νkv(t)k2H˙ β+α/2 ≤ ku(t)kH˙ α/2 kv(t)kH˙ β+α/2 . 2 dt The previous inequality implies Z t kv(s)k2H˙ β+α/2 ds kv(t)k2H˙ β/2+α/2 + ν 0 Z t 2 kΛα/2 u(s)k2L2 ≤ kv0 k2H˙ β/2+α/2 + C ≤ kv0 kH˙ β/2+α/2 + c(ν) 0

with constant C = C(α, β, µ, ν, λ, ku0 kL1 (T) , ku0 kL2 (T) , kv0 kH β−α/2 (T) , T ).


22

We compute d ku(t)k2H˙ α/2 dt

Z

Z

Z

= −µ |Λ u| dx + uΛ vΛ udx + r |Λα/2 u|2 dx T T Z Z T α/2 2 β−1 α HvΛ udx − r Λ (u )Λα/2 udx + ∂x uΛ α

2

β

α

T

T

3µ ≤ − ku(t)k2H˙ α + c(µ)ku(t)k2L2 kΛβ v(t)k2L∞ 4 +rku(t)k2H˙ α/2 + kukH˙ 1 kΛβ−1 HvkL∞ kukH˙ α +c(r)kΛα/2 u(t)kL∞ kukL2 ku(t)kH˙ α/2 .

We use the interpolation inequalities (α−1)/α

1/α

kukH˙ 1 ≤ ckukH˙ α kukL2

,

kΛα/2 ukL∞ ≤ ckukH˙ α ,

kΛβ v(t)k2L∞ ≤ ckv(t)k2H˙ β+α/2 ,

kΛβ−1 HvkL∞ ≤ ckvkH˙ β−1+α/2 ≤ ckvkH˙ β/2+α/2 , to obtain ku(t)k2H˙ α/2

µ + 2

Z

t 0

ku(t)k2H˙ α ≤ ku0 k2H˙ α/2 + C,

where the constant depends on C = C(α, β, µ, ν, λ, ku0 kL1 (T) , ku0 kL2 (T) , kv0 kH β−α/2 (T) , kv0 kH β/2+α/2 (T) , T ). Notice that, in the case α > 1.5, we have Z

T 0

k∂x u(t)kL∞ + kΛv(t)kL∞ dt ≤ c

Z

T 0

ku(t)kH α + kv(t)kH β+α/2 dt ≤ C,

so, in this case, we are done with the entire proof. Step 3. (a priori estimates III) In this step we obtain that u ∈ L∞ (0, T ; H α (T)) ∩ L2 (0, T ; H 3α/2 (T)),

v ∈ L∞ (0, T ; H β/2+α (T)) ∩ L2 (0, T ; H β+α (T)).

Testing the equation for v against Λβ+2α v, we obtain kv(t)k2H˙ β/2+α

+ν

Z

t 0

kv(s)k2H˙ β+α ds ≤ kv0 k2H˙ β/2+α + C.


23

We compute Z Z Z d β 2α 3α/2 2 2 u| dx + uΛ vΛ udx + r |Λα u|2 dx ku(t)kH˙ α = −µ |Λ dt T T T Z Z + ∂x uΛβ−1 HvΛ2α udx − r Λα (u2 )Λα udx T

T

≤ −µku(t)k2H˙ 3α/2 + kΛα/2 (uΛβ v)kL2 ku(t)kH˙ 3α/2

+rku(t)k2H˙ α + kΛα/2 (∂x uΛβ−1 Hv)kL2 ku(t)kH˙ 3α/2

+cku(t)kL∞ ku(t)k2H˙ α µ ≤ − ku(t)k2H˙ 3α/2 + c(µ)[ku(t)k2H˙ α/2 kΛβ vk2L∞ 2 +ku(t)k2L∞ kΛβ+α/2 vk2L2 ] + rku(t)k2H˙ α

+c(µ)[ku(t)k2H˙ 1+α/2 kΛβ−1 Hvk2L∞ + k∂x uk2L2 kΛβ−1+α/2 vk2L∞ ]

+cku(t)kL∞ ku(t)k2H˙ α .

Step 4. (construction of a solution) If the initial data (u0 , v0 ) ∈ H kα (T) × H kα+β/2 (T), k ∈ N, kα ≥ 3 we have local existence of regular solutions from Theorem 1. Additionally, Step 3 gives us bounds u ∈ L∞ (0, T ; H α (T)) ∩ L2 (0, T ; H 3α/2 (T)),

v ∈ L∞ (0, T ; H β/2+α (T)) ∩ L2 (0, T ; H β+α (T))

that are independent from the local time of existence; let us call them globalin-time bounds. In fact, to obtain the global-in-time bounds rigorously, using regularity given by Theorem 1, we need at step 3 to reinterpret some intermediate steps in terms of duality pairing. A similar remark applies to obtaining the evolutionary norms in all steps 1-3, including the notion of the time derivative of a single (spatial) Fourier mode. This point has been already raised by the end of Section 5. Our global-in-time bounds give Z T k∂x u(s)kL∞ ds ≤ C(T ). 0

To conclude with the continuation criterion given by Theorem 2, we need also Z T kΛβ v(s)kL∞ ds ≤ C(T ). 0

In fact, using β − 1 + α/2 ≤ β/2 + α/2 we obtain

kΛβ−1+α/2 v(t)k2L∞ , kΛβ−1 Hv(t)k2L∞ ∈ L∞ , kΛβ+α/2 v(t)k2L2 , kΛβ v(t)k2L∞ ∈ L1

Next let us consider the case where the initial data is not that smooth, but merely (u0 , v0 ) ∈ L2 × H β−α/2 . After mollification, we have an initial data (uǫ0 , v0ǫ ) with the desired regularity. Applying the previous reasoning, we have a global smooth regularized solution (uǫ , v ǫ ). Due to Step 1, these functions are uniformly bounded in uǫ ∈ L∞ ([0, T ], L2 (T)) ∩ L2 ([0, T ], H α/2 (T)),

v ǫ ∈ L∞ ([0, T ], H β−α/2 (T)) ∩ L2 ([0, T ], H 3β/2−α/2 (T)).


24

Testing ∂t uǫ , ∂t v ǫ against φ ∈ H 2 and using the duality pairing, we obtain a uniform bound ∂t uǫ , ∂t v ǫ ∈ L∞ ([0, T ], H −2 (T)).

Applying Aubin-Lions’s Theorem (with H α/2 ⊂ L2 ⊂ H −2 for uǫ and H 3β/2−α/2 ⊂ H β−α/2 ⊂ H −2 for v ǫ ), we take a subsequence (denoted again by ǫ) such that uǫ (t) → u(t) in L2t L2x ,

uǫ (t) ⇀ u(t) in L2t Hxα/2 ,

v ǫ (t) → v(t) in L2t Hxβ−α/2 , v ǫ (t) ⇀ v(t) in L2t Hx3β/2−α/2 . Using the properties of the mollifier, we have uǫ (0) → u0 in L2 , v ǫ (0) → v0 in L2 .

With the previous strong convergence, we can pass to the limit in the weak formulations of Definition 1. Step 5. (absorbing set) We write !2 X |k|β−α/2 (24) CF S (β, α, λ, ν) = ν|k|β + λ k∈Z

According to (23), for every t ≥ 0, we have so, Z

kv(t)k2H˙ β−α/2 ≤ kv0 k2H˙ β−α/2 e−λt + N CF S (β, α, λ, ν),

t+1 t

kv0 k2H˙ β−α/2

kv(s)k2H˙ β−α/2 ds ≤

Z

t

λ

kv0 k2H˙ β−α/2

≤ t+1

λ

(1 − e−λ )e−λt + N CF S (β, α, λ, ν) + N CF S (β, α, λ, ν),

2+2δ 2+2δ 2δ δ . ds ≤ kv0 k2H˙ β−α/2 + N CF S (β, α, λ, ν) kv(s)kH˙ β−α/2

Notice that if

 

 1 N CF S (β, α, λ, ν)  t ≥ t0 = max 0, , log  kv k2 0 ˙ β−α/2  −λ  H −λ (1 − e ) λ 

we have an inequality that is independent of v0 :

kv(t)k2H˙ β−α/2 ≤ 2N CF S (β, α, λ, ν). Then, from (21), we obtain µ (CKP (α)N )2 d ku(t)k2L2 + ku(t)k2H˙ α/2 ≤ rku(t)k2L2 + kv(t)k2H˙ β−α/2 dt 2 µ + Due to Lemma 2, we obtain Z

t

t+1

(CKP (α)CGN (α))

ku(s)k2L2 ≤ 3N .

2+2δ δ

µ

2+2δ

δ 4N 2 kv(t)kH˙ β−α/2

.


25

Using Uniform Gronwall estimate (Lemma 5) and the previous inequality, we have that ku(t + 1)k2L2 ≤ S(L2 ) ∀ t ≥ t0 ,

where S(L2 ) is defined in (40). Using the previous inequality we also obtain Z t+1 2(S(L2 ) + S(L2 )e−r ) ku(s)k2H˙ α/2 ds ≤ , ∀ t ≥ t0 + 1. µ t Let us consider α < 2 (the case α = 2 can be done straightforwardly). We look for a commutator-type structure in the nonlinearity: Z Z 1 d α 2 2 ku(t)kH˙ α/2 = −µ |Λ u| dx + uΛβ vΛα udx 2 dt T Z ZT α/2 2 +r |Λ u| dx − r u2 Λα udx T Z Th i Λα/2 , Λβ−1 Hv ∂x uΛα/2 udx + T

+

Z

Λβ−1 Hv

T

We estimate I1 =

Z

uΛβ vΛα udx ≤

T

I2 = r

Z

T

I3 =

Z

T

u2 Λα udx ≤

Λβ−1 Hv

∂x (Λα/2 u)2 dx. 2

2 µ kuk2L∞ kvk2H˙ β + kuk2H˙ α , µ 8

µ 2r 2 kuk2L∞ kuk2L2 + kuk2H˙ α , µ 8

∂x (Λα/2 u)2 dx ≤ CI (α)kvkH˙ β kukH˙ α/2 kukH˙ α , 2

where we have kf k2L4 ≤ CI (α)kf kL2 kf kH˙ α/2 .

(25)

The yet untouched term is Z h i Λα/2 , Λβ−1 Hv ∂x uΛα/2 udx I4 = T

h i

≤ Λα/2 , Λβ−1 Hv ∂x u

L2

kukH˙ α/2

We use the inequalities (10) together with the following Kenig-Ponce-Vega estimate (see Lemma 4) k[Λα/2 , Λβ−1 Hv]∂x ukL2 ≤ CKP V (α) k∂x uk 2+ 2α−2 kΛβ−1+α/2 HvkL2/(α−1) 2−α L (26) +kukW α/2,∞ kvkH˙ β , and we obtain

3 1 4 I4 ≤ CKP V (α)kukH˙ α/2 kukH˙ α kvkH˙ β CSE (α)CSE (α) + CSE (α) ,

by Young’s inequality it yields 3 (α)C 1 (α) + C 4 (α)) 2 CKP V (α)(CSE µ SE SE 2 2 I4 ≤ kvkH˙ β kukH˙ α/2 + kuk2H˙ α . µ 4


26

Collecting every estimate, we have that d kuk2H˙ α/2 + µkuk2H˙ α ≤ 2kuk2H˙ α/2 g(t), dt with g(t) = r +

2 (α)r 2 kvkH˙ β 2CSE kuk2L2 + µ 2 2 2CSE (α) + (CI (α))2 + kvk2H˙ β µ 3 (α)C 1 (α) + C 4 (α)) 2 CKP V (α)(CSE SE SE kvk2H˙ β . + µ

Testing the equation for v with Λβ v and using β ≥ α, we have Z t+1 N 3 2 + 2CF S (β, α, λ, ν) ∀ t ≥ t0 , kv(s)kH˙ β ds ≤ ν ν t

so, if t ≥ t0 , Z t+1 2 (α) r 2 CSE N 3 g(s)ds ≤ r + 6N + + 2CF S (β, α, λ, ν) µ ν ν t 2 (α) + (C (α))2 1 2CSE I × + 2 µ ! 3 (α)C 1 (α) + C 4 (α)) 2 CKP V (α)(CSE SE SE . × µ Using Lemma 5, we have that

with

ku(t + 1)k2H˙ α/2 ≤ S(H˙ α/2 ) ∀ t ≥ t0 + 1, Z t+1 Z t+1 1 2 + g(s)ds ∀ t ≥ t0 + 2, ku(s)k2H˙ α ds ≤ S(H˙ α/2 ) µ 2 t t 2(S(L2 ) + S(L2 )e−r ) 2 R t+1 g(s)ds . e t S(H˙ α/2 ) = µ

Hence we have obtained the absorbing set in H kα with k = 1. Due to the linear character of the equation for v, we have that kv(t)k2H˙ β ≤ kv0 k2H˙ β e−λt + M(H˙ α/2 )CF S (β, α, λ, ν), where, for a given space X, we set √ M(X) = 2π max max ∗ ku(s)kX , S(X) , 0≤s≤T

for T ∗ >> 1 that will be fixed later. We remark that kΛα/2 u(t)kL1 ≤ M(H˙ α/2 ). Now we can continue in the same way using induction. Once we have the absorbing set for u in L∞ ([0, ∞], H kα/2 (T)) (k ≥ 1) and the bound u ∈ L2 ([t, t+1], H (k+1)α/2 (T)), we can ensure that v ∈ L∞ ([0, ∞], H β+(k−1)α/2 (T))


27

and v ∈ L2 ([t, t + 1], H β+kα/2 (T)). Now we test the equation for u against Λ(k+1)α u to get Z Z k 1 d ku(t)k2H˙ (k+1)α/2 = −µ |Λ( 2 +1)α u|2 dx + r |Λ(k+1)α/2 u|2 dx 2 dt T Z T kα/2 β (k/2+1)α (uΛ v)Λ udx + Λ ZT −r Λkα/2 (u2 )Λ(k/2+1)α udx Z Th i Λ(k+1)α/2 , Λβ−1 Hv ∂x uΛ(k+1)α/2 udx + Z

T

∂x (Λ(k+1)α/2 u)2 + Λβ−1 Hv dx. 2 T To conclude the existence of S (k+1)α we use Lemma 4 and the same ideas. 2 Finally notice that at each iteration step we have to add 1 to the initial value t0 . Consequently, we can take T ∗ = T ∗ (k) large enough to reach H kα (T). For instance, to reach H 3 , T ∗ = t0 + 10 is enough. 9. Proof of Theorem 5: Smoothing effect

Here we prove our main result concerning the smoothing effect of the system (1)-(2): Proof of Theorem 5. We recall the Hardy-Sobolev norm (14). Recall that for t > 0 the finiteness of this norm implies the analyticity on the real line. We define z = x ± iωt. In this complex strip the extended system is ∂t u(z) = −µΛα u(z) + ∂x · (u(z)Λβ−1 Hv(z)) +ru(z)(1 − u(z)),

(27)

∂t v(z) = −νΛβ v(z) − λv(z) + u(z).

(28)

We are going to perform new energy estimates in the Hardy-Sobolev space (13) for an appropriate value of ω. Notice that, as the functions u and v are complex for complex arguments, the integration by parts is a delicate matter for some terms. Consequently there are several new terms appearing that are not present in the real case. We deal first with the case α, β > 1. At the end of the proof we explain how to cover the extreme case α = β = 1. We restrict here to formal estimates, as their rigorization is analogous to that for the real case. R R Let us start with the estimates for the equation (28). Using f g¯ = f¯g, we have Z d 2 kvkL2 (Sω ) = 2Re v¯(z) (∂t v(z) ± iω∂x v(z)) dx. dt T Using Plancherel’s Theorem, we have Z Re v¯(z)(−νΛβ v(z) − λv(z))dx = −νkvk2H˙ β/2 (S ) − λkvk2L2 (Sω ) ≤ 0. T

ω

Consequently, using (28), 1 d kvk2L2 (Sω ) ≤ kvkL2 (Sω ) ωkvkH 1 (Sω ) + kukL2 (Sω ) . 2 dt


28

Taking 4 derivatives of the equation (28) and testing against ∂x4 v, we obtain Z 1 d kvk2H˙ 4 (S ) = Re ∂x4 v¯(z) ∂t ∂x4 v(z) ± iω∂x5 v(z) dx ≤ I1 + I2 + I3 , ω 2 dt T

with

I1 = −νRe

Z

T

∂x4 v¯(z)∂x4 Λβ v(z)dx = −νkvk2H˙ 4+β/2 (S ) , ω

I2 = −Re = −Re

Z

ZT

T

∂x4 v¯(z)ΛH∂x3 u(z)dx Λ0.5 ∂x4 v¯(z)Λ0.5 H∂x3 u(z)dx

≤ 2kvkH˙ 4+1/2 (Sω ) kukH˙ 3+1/2 (Sω ) , I3 = ∓ωRe

Z

T

∂x4 v¯(z)ΛH∂x4 v(z)dx ≤ 2ωkvk2H˙ 4.5 (S ) . ω

Using Young’s inequality and the interpolation inequality (α−1)/α

kf kH 0.5 ≤ CSI (α)kf kL2

1/α

kf kH˙ α/2 ,

we have that (CSI (α))2 2/α 2(α−1)/α kukH˙ 3 (S ) kukH˙ 3+α/2 (S ) ω ω ω ω ! 2 2 kukH˙ 3 (S ) kuk2H˙ 3+α/2 (S ) (C (α)) SI ω ω α ≤ ωkvk2H˙ 4+1/2 (S ) + , α α/(α−1) + ǫ ω ω α ǫ α−1

I2 ≤ ωkvk2H˙ 4+1/2 (S

+ )

for every ǫ > 0. Now we fix ǫ=

ωα µ 4 (CSI (α))2

1/α

to conclude µ kuk2H˙ 3+α/2 (S ) ω 4 −1/(α−1) µ 2 (CSI (α)) α − 1 4 ωα + kuk2H˙ 3 (S ) . ω ω α (CSI (α))2

I2 ≤ ωkvk2H˙ 4+1/2 (S

ω)

+

Collecting all the estimates for v we obtain µ 1 d kvk2H 4 (Sω ) ≤ −νkvk2H˙ 4+β/2 (S ) + 3ωkvk2H˙ 4+1/2 (S ) + kuk2H˙ 3+α/2 (S ) ω ω ω 2 dt 4 −1/(α−1) µ 2 (CSI (α)) α − 1 4 ωα + kuk2H˙ 3 (S ) ω ω α (CSI (α))2 2 kukH 3 (Sω ) 1 . kvk2H 4 (Sω ) + + ω+ 2 2


29

Now we proceed with the equation for u. The lower order term can be bounded easily as follows Z 1 d ¯(z) (∂t u(z) ± iω∂x u(z)) dx kuk2L2 (Sω ) = Re u 2 dt T ≤ kukL2 (Sω ) ωkukH 1 (Sω ) + kukL∞ (Sω ) kvkH˙ β (Sω ) + rkukL2 (Sω ) + rkukL2 (Sω ) kukL∞ (Sω ) +kukH 1 (Sω ) kΛβ−1 HvkL∞ (Sω ) .

The higher order seminorm contributes with Z 1 d 2 ¯(z) ∂t ∂x3 u(z) ± iω∂x4 u(z) dx = I4 + I5 + I6 + I7 , kukH˙ 3 (S ) = Re ∂x3 u ω 2 dt T with Z ¯(z)∂x3 Λα u(z)dx = −µkuk2H˙ 3+α/2 (S ) , I4 = −µRe ∂x3 u ω T Z ¯(z)ΛH∂x3 u(z)dx ≤ 2ωkuk2H˙ 3.5 (S ) , I5 = ∓ωRe ∂x3 u ω

T

I6 = rRe

Z

T

and I7 = Re with

Z

T

∂x3 u ¯(z)∂x3 [u(z)(1 − u(z))] dx ≤ rkuk2H˙ 3 (S

ω)

1 + 2kukL∞ (Sω ) + 6k∂x ukL∞ (Sω ) ,

h i ∂x3 u ¯(z)∂x4 u(z)Λβ−1 Hv(z) dx = K1 + K2 + K3 + K4 + K5 .

Z

¯(z)u(z)∂x3 Λβ v(z)dx ≤ kukH˙ 3 (Sω ) kukL∞ (Sω ) kvkH˙ 3+β (Sω ) , ∂x3 u T Z ¯(z)∂x3 u(z)Λβ v(z)dx ≤ 4kuk2H˙ 3 (S ) kΛβ vkL∞ (Sω ) , K2 = 4Re ∂x3 u ω T Z ¯(z)∂x2 u(z)∂x Λβ v(z)dx ≤ 6kuk2H˙ 3 (S ) kΛβ ∂x vkL∞ (Sω ) , K3 = 6Re ∂x3 u ω T Z ¯(z)∂x u(z)∂x2 Λβ v(z)dx ≤ 4k∂x ukL∞ (Sω ) kukH˙ 3 (Sω ) kvkH˙ 2+β (Sω ) , K4 = 4Re ∂x3 u T Z ¯(z)∂x4 u(z)Λβ−1 Hv(z)dx. K5 = Re ∂x3 u K1 = Re

T

Splitting the real and imaginary parts, we have

K5 = L1 + L2 + L3 , with 1 L1 = Re 2

L3 = −

Z

Z

1 Λβ−1 Hv(z)∂x |∂x3 u(z)|2 dx ≤ kuk2H˙ 3 (S ) kΛβ vkL∞ (Sω ) , ω 2 T Z ImΛβ−1 Hv(z)∂x3 Re u(z)∂x4 Im u(z)dx, L2 = T

T

ImΛβ−1 Hv(z)∂x3 Im u(z)∂x4 Re u(z)dx ≤ L2 +kuk2H˙ 3 (S ) kΛβ vkL∞ (Sω ) . ω


30

Using ΛH = −∂x and the self-adjointness of Λs , we find a commutator

h i

kukH˙ 3.5 (Sω ) L2 ≤ Λ0.5 , ImΛβ−1 Hv ∂x3 Re u 2 L (Sω )

+ kImΛβ−1 HvkL∞ (Sω ) kuk2H˙ 3.5 (S ) . ω

We use Lemma 4 to obtain the commutator estimate

h i

0.5 2 β β−1 3 ≤ CKP Λ , ImΛ Hv ∂ Re u

2

V kΛ vkL∞ (Sω ) kukH˙ 3 (Sω ) . x L (Sω )

Putting all the estimates together and using Sobolev embedding and 3+β ≤ 4 + β/2, we have that d 2 (α)kvkH˙ 4 (Sω ) + 2r kuk2H 3 (Sω ) ≤ 2kuk2H 3 (Sω ) ω + 17.5CSE dt 2 2 +2CSE (α)kvkH˙ 4+β/2 (Sω ) + kuk3H 3 (Sω ) 9rCSE (α) −2µkuk2H˙ 3+α/2 (S

+ 4ωkuk2H˙ 3.5 (S

ω) 2 +2kIm Λ HvkL∞ (Sω ) kukH˙ 3.5 (S ) ω 2 2 +2kukH˙ 3.5 (Sω ) kukH˙ 3 (Sω ) CKP V CSE (α)kvkH 4 (Sω ) . ω)

β−1

Notice that, using Poincaré inequality √ √ kIm Λβ−1 HvkL∞ ≤ 2kIm vkH˙ β ≤ 2kIm vkH˙ 2 . We define the energy 1 − 2kIm vkH˙ 2

1 1

.

+ + 2ku0 kL∞ (T) − |u(z)|2 L∞ 2kv0 kL∞ (T) − |v(z)|2 L∞

E(t) = 1 + kuk2H 3 (Sω ) + kvk2H 4 (Sω ) +

We have d dt

µ 4

µ 4

√

√ d 1 E(t) √ ≤ 2(E(t))2 kIm vkH˙ 2 ≤ (E(t))4 + , dt 2 − 2kIm vkH˙ 2

and 1 1 d ≤ E(t)kukL∞ k∂t ukL∞ . dt 2ku0 kL∞ (T)−|u(z)|2 2ku0 kL∞ (T)−|u(z)|2 Integrating this ODI, taking the absolute value and using the limit definition for the derivative, we obtain (see [20, 24, 25] for further details)

1 d 2

≤ (µ + 2 + 2r)CSE (α)(E(t))4 .

2 dt 2ku0 kL∞ (T) − |u(z)| L∞

In the same way

1 d 2

≤ (ν + λ + 1)CSE (α)(E(t))3 .

2 dt 2kv0 kL∞ (T) − |v(z)| L∞


31

Thus, putting all the estimates together, we get d µ E(t) ≤ −2νkvk2H˙ 4+β/2 (S ) + 6ωkvk2H˙ 4+1/2 (S ) + 2( − µ)kuk2H˙ 3+α/2 (S ) ω ω ω dt 4 −1/(α−1) 2 2 (CSI (α)) α − 1 ω α +2 + 2ω ω α (CSI (α))2 2 (α))2 (9rCSE 2 2 +1.25 + 2r + (17.5CSE (α)) + E(t) 2 2 2 2 2(CKP 2 V CSE (α)) +(E(t)) 1.5 + µ

2 +(1 + (µ + 2 + 2r + ν + λ + 1)CSE (α))(E(t))4 µ √ 3 + + ( 2) kIm vkH˙ 2 + 4ω kuk2H˙ 3.5 (S ) . ω 2 √ Now observe that, as long as E(t) < ∞, we have µ4 − 2kIm vkH˙ 2 > 0 and, using Poincaré inequality if needed, we obtain

d µ E(t) ≤ −2νkvk2H˙ 4+β/2 (S ) + 6ωkvk2H˙ 4+1/2 (S ) − kuk2H˙ 3+α/2 (S ) ω ω ω dt 2 −1/(α−1) ω2α (CSI (α))2 α − 1 + 2ω +2 ω α (CSI (α))2 2 (α))2 (9rCSE 2 2 +1.25 + 2r + (17.5CSE (α)) + E(t) 2 2 2 2 2(CKP 2 V CSE (α)) +(E(t)) 1.5 + µ 2 +(1 + (µ + 2 + 2r + ν + λ + 1)CSE (α))(E(t))4

+4ωkuk2H˙ 3.5 (S ) . ω

For α, β > 1 we have by Plancharel Theorem µ 4ωkuk2H˙ 3.5 (S ) − kuk2H˙ 3+α/2 (S ) ≤ C1 kuk2H˙ 3 (S ) , ω ω ω 2 3ωkvk2H˙ 4.5 (S

ω)

− νkvk2H˙ 4+β/2 (S

ω)

≤ C2 kvk2H˙ 4 (S ) , ω

with C1 , C2 given by (46),(47). Consequently, we can choose any positive value for ω > 0 and we have the inequality d E(t) ≤ K1 (E(t))4 , dt with K1 , Ci defined in (46),(47) and (43). Solving this ODI, we obtain E(t) ≤ r 3

1

1

1+ku0 k2 3 +kv0 k2 4 H (T) H (T)

, − 3tK1

and, using (43), we conclude that (u, v) are analytic functions at least for time 1 T˜ = 2 3K1 (1 + ku0 kH 3 (T) + kv0 k2H 4 (T) )


32

Notice that in the extreme cases min{α, β} = 1, we can take 0 < ω ≤ ω0 , (with ω0 defined in (11)) to obtain the inequality d E(t) ≤ K2 (E(t))4 , dt with K2 given by (44). From the inequality (29) we obtain

(29)

E(t) ≤ r 3

1

1+ku0 k2

1

H 3 (T)

,

+kv0 k2

H 4 (T)

− 3tK2

and we again conclude that the solution (u, v) is analytic for time t < T˜ with 1 . T˜ = 3K2 (1 + ku0 k2H 3 (T) + kv0 k2H 4 (T) ) Remark 4. A similar Theorem holds for the parabolic-elliptic system ( (1)(2) with τ = 0). We refer the reader to [1] for details on how to adapt the proof. Proof of Corollary 1. The proof of Corollary 1 is obtained by a standard continuation argument. First notice that the solution (u(t), v(t)) ∈ H 3 (T) × H 4 (T) globally and it is unique. In particular, at t = T˜, we can restart the evolution with initial data (u10 , v01 ) = (u(T˜), v(T˜)). The initial data may not be analytic, but there exists a δ small enough so (u1 (t), v 1 (t)) = (u(T˜ + t), v(T˜ + t)) is analytic for 0 < t < δ. As we can find such a positive δ for every initial data, we conclude. In other words, if we can not find such a positive δ, it is because (un0 , v0n ) ∈ / H 3 (T) × H 4 (T), and this is a contradiction. For the proof of Corollary 2 we refer to [1, 20, 24]. 10. Proof of Theorem 6: Existence of the attractor Here we prove the existence and some properties of the attractor. First we need a definition from dynamical systems (see [45]). Definition 3. The solution operator S(t)(u0 , v0 ) = (u(t, x), v(t, x)) defines a compact semiflow in H 3α (T)×H β/2+3α (T) if, for every (u0 , v0 ) ∈ H 3α (T)× H β/2+3α (T) the following statements hold: • S(0)(u0 , v0 ) = (u0 , v0 ). • for all t, s, u0 , v0 , the semigroup property hold, i.e., S(t + s)(u0 , v0 ) = S(t)S(s)(u0 , v0 ) = S(s)S(t)(u0 , v0 ). • For every t > 0,

S(t)(·, ·) : H 3α (T) × H β/2+3α (T) 7→ H 3α (T) × H β/2+3α (T)

is continuous.


33

• There exists T ∗ > 0 such that S(T ∗ ) is a compact operator, i.e. for every bounded set B ⊂ H 3α (T) × H β/2+3α (T), S(T ∗ )B ⊂ H 3α (T) × H β/2+3α (T) is a compact set. We have Lemma 3. Given T > 0, 8/7 ≤ α ≤ β ≤ 2, µ, ν, λ, r > 0, (u0 , v0 ) ∈ H 3α × H β/2+3α , then S(·)(u0 , v0 ) = (u(·)), v(·))) ∈ C([0, T ], H 3α (T) × H β/2+3α (T)) for every initial data and it defines a compact semiflow in H 3α × H β/2+3α . Proof. As in Theorem 4 we have (30)

Λ3α u ∈ L∞ ([0, T ], L2 ) ∩ L2 ([0, T ], H α/2 ),

(31)

Λ3α+β/2 v ∈ L∞ ([0, T ], L2 ) ∩ L2 ([0, T ], H β/2 ).

We have to prove that

∂t Λ3α u ∈ L2 ([0, T ], H −α/2 ), ∂t Λ3α+β/2 v ∈ L2 ([0, T ], H −β/2 ).

By duality and Kato-Ponce inequality, we have Z 3α 3α k∂t Λ ukH˙ −α/2 = sup ∂t Λ uφdx kφkH α/2 ≤1 3.5α

≤ µkΛ

T

ukL2 + kΛ2.5α (uΛβ v)kL2 + kΛ2.5α (∂x uΛβ−1 Hv)kL2

+rkΛ2.5α ukL2 + rkΛ2.5α (u2 )kL2 ≤ C kΛ3.5α ukL2 + kΛ2.5α ukL2 kΛβ vkL∞ + kΛβ+2.5α vkL2 kukL∞ +kΛ2.5α ∂x ukL2 kΛβ−1 HvkL∞ + k∂x ukL∞ kΛβ+2.5α−1 vkL2 +kΛ2.5α ukL2 + kukL∞ kΛ2.5α ukL2

≤ C kukH˙ 3.5α + kukH 3α kvkH 3α+β/2 + kvkH˙ β+3α kukH 3α +kukH˙ 3.5α kvkH β/2+3α + kukH 3α kvkH β+3α +kukH 3α + kukH 3α kukH 3α ) ,

and we conclude the bound for ∂t Λ3α u: Z T k∂t Λ3α u(s)k2H˙ −α/2 ds ≤ C (32) 0

We proceed in the same way to get Z T k∂t Λ3α+β/2 v(s)k2H˙ −α/2 ds ≤ C (33) 0

The bounds (32) and (33) together with (30) and (31) imply u ∈ C([0, T ], H˙ 3α ), v ∈ C([0, T ], H˙ 3α+β/2 )

To get the full norm we use

u ∈ L∞ ([0, T ], L2 ) ∩ L2 ([0, T ], H α/2 ), v ∈ L∞ ([0, T ], L2 ) ∩ L2 ([0, T ], H β/2 ).

and repeat the argument for This implies

∂t u ∈ L2 ([0, T ], H −α/2 ), ∂t v ∈ L2 ([0, T ], H −β/2 ). S(·)(u0 , v0 ) ∈ C([0, T ], H 3α × H β/2+3α ).


34

The semigroup property follows from the uniqueness of the classical solution. Fixed s0 , the continuity of S(s0 )(·, ·) : H 3α × H β/2+3α 7→ H 3α × H β/2+3α ,

can be obtained with the energy estimates. Finally, we use Theorem 5 to get that S(t)(u0 , v0 ) ∈ H 3.5α × H β+3α if t ≥ δ, for every initial data and δ > 0. As in Theorem 4, we obtain the existence of T ∗ and a constant C such that max∗ {ku(t)kH 3.5α + kv(t)kH β+3α } ≤ C. t≥T

Using the compactness of the embeddings H ǫ ֒→ L2 , we conclude the result. Remark 5. The restriction α ≤ β is to get the existence of the absorbing sets when applying Theorem 4. The restriction 8/7 ≤ α is to get to invoke Theorem 5.

3α + β/2 ≥ 3.5α ≥ 4,

Proof of Theorem 6. We can use the previous Lemma together with Theo rem 4 and Theorem 1.1 in [45] to conclude Theorem 6. 11. Proof of Theorem 7: The number of relative maxima Finally, we provide the proof of Theorem 7: Proof of Theorem 7. Using Theorem 5 for T˜/(N − 1) < t < T˜, N ≥ 3 and ω = ω0 defined in (11), we have that the solutions become analytic in a strip with width at least 2 2 ω T˜ ω 1 + ku0 kH 3 (T) + kv0 kH 4 (T) W= = , N N 3K and K given by (45). We have 1 1 ˜ − ωT ≤ ωt − W N −1 N and using Cauchy’s formula and Hadamard’s three lines theorem, √ N (N − 1)kukL∞ ({|ℑz|≤ωt}) 2(N − 1)ku0 kL∞ (T) , k∂x ukL∞ ({|ℑz|≤W}) ≤ ≤ W ω T˜ √ N (N − 1)kvkL∞ ({|ℑz|≤ωt}) 2(N − 1)kv0 kL∞ (T) k∂x vkL∞ ({|ℑz|≤W}) ≤ . ≤ W ω T˜ Using Lemma 6, we have that for any ǫ > 0, 0 < T˜/(N − 1) < t < T˜, T = Iǫu ∪ Rǫu = Iǫv ∪ Rǫv , where Iǫu , Iǫv are the union of at most [ 4π W ] intervals open in T, and • |∂x u(x)| ≤ ǫ, for all x ∈ Iǫu , • card{x ∈ Rǫu : ∂x u(x) = 0} ≤ • |∂x v(x)| ≤ ǫ, for all x ∈ Iǫu ,

• card{x ∈

Rǫv

: ∂x v(x) = 0} ≤

2 2π log 2 W

2 2π log 2 W

log

log

√ 2(N −1)ku0 kL∞ (T) Wǫ

√

2(N −1)kv0 kL∞ (T) Wǫ

,

.


35

Proof of Corollary 3. Notice that in the case min{α, β} > 1, we are free to choose N , ω= 2 1 + ku0 kH 3 (T) + kv0 k2H 4 (T) and we can improve the statement in Theorem 7. Applying Lemma 6, we have that for any ǫ > 0, T = Iǫu ∪ Rǫu = Iǫv ∪ Rǫv , with Iǫu , Iǫv are the union of at most [12πK1 ] intervals open in T, and • |∂x u(x)| ≤ ǫ, for all x ∈ Iǫu ,

• card{x ∈ Rǫu : ∂x u(x) = 0} ≤

• |∂x v(x)| ≤ ǫ, for all x ∈ Iǫu , • card{x ∈

Rǫv

: ∂x v(x) = 0} ≤

12πK1 log 2

12πK1 log 2

log

log

√

√

18K1 (N −1)ku0 kL∞ (T) ǫ

18K1 (N −1)kv0 kL∞ (T) ǫ

,

.

We are interested in the points of maximum such that they are close to regions with derivative bigger than one (the so-called peaks). Consequently, we take ǫ = 1 and N = 3. Finally, notice that, in the attractor, we have 2 ku(t)kL∞ ≤ CSE (α)S(H α/2 )

to conclude the result.

The proof of Corollary 4 follows from the same ideas as before. 12. Numerical study 12.1. Algorithm. The dynamics differs substantially depending on the value of the parameters presents in the problem. Hence, in order to reduce the number of parameters, let us consider the one-parameter problem (34) (35)

∂t u = −Λα u + χ∂x · (uΛβ−1 Hv) + u(1 − u) ∂t v = −Λβ v − v + u,

with χ > 0. This new system can be related to (1)-(2) by the appropriate choice of scaling. To simulate this problem we use a well-known Fourier-collocation method. First we discretize the spatial domain using N uniformly distributed points. Notice that the numerical solution (uN , vN ) will have N points for uN and N points for vN . We use the Fast Fourier Transform (FFT) to change to the frequency space. There the differential operators and the Hilbert transform act as multipliers. Indeed, if we denote the FFT using FFT(·), we have FFT(Λγ uN ) = |ξ|γ FFT(uN ), FFT(Λβ−1 HuN ) = −iξ|ξ|β−2 FFT(uN ). To compute the nonlinear term we use the Inverse Fast Fourier Transform (IFFT) to change back to the physical space. We multiply there appropriately and then we go back to the frequency space using FFT. In particular, writing IFFT(·) for the IFFT we have that the nonlinearity can be written as iξFFT uN IFFT(−iξ|ξ|β−2 uˆN ) .


36

Emerging with α=β=1,χ=20 9

t=36.9 t=37.1 8

t=37.3 7

u(x,t)

6

5

4

3

2

1

0

−3

−2

−1

0

1

2

3

x

Figure 3. Emerging peak. In this way we can write our problem as a ordinary differential equation in the frequency space. Now we can advance up to time T using our favorite numerical integrator. In particular, we choose the function ode45 in Matlab. 12.2. Results. 12.2.1. α = 1, β = 1. First, we study the case α = 1 and β = 1. For high values of χ, we observe the same chaotic behavior as in [41]. The homogeneous steady state is unstable and a number of peaks eventually We take emerge and merge with other peaks (see Figures 3, 4 and 5). N = 213 , T = 30 and the initial data (36)

u0 (x) = 1, v0 (x) = 0.1 sin(8x) + 1.

In order to better understand the role of χ, we approximate the solution for different values χi ∈ [5, 20]. The step between our values is χi+1 − χi = 0.5. The outcome is plotted in Figure 6. In the part a of the figure, we plot the solutions corresponding to different values of χ and times 20 ≤ t ≤ 30. Notice that every line corresponds to a fixed time t. We see that for lower values of χ the solution tends to the homogeneous steady state, while for large values of χ the solution develops chaotic behavior. In particular, we can see how a small change in χ has a big impact on the solution at a fixed time ti . We can see also how, for a fixed value χi , the solution at different times take very different values. In part b of the same figure, we plot kuN (t)kL∞ for χ = 5 (solid line) and χ = 20 (dotted line).


37

Merging with α=β=1,χ=20 22

t=34.4 20

t=34.5 18

t=34.6 16

14

u(x,t)

12

10

8

6

4

2

0

−3

−2

−1

0

1

2

3

x

Figure 4. Merging peaks.

Figure 5. Tracking the peaks for the case α = β = 1, χ = 20 up to time T = 100. 12.2.2. α = 1.5, β = 2. Now we study the case α = 1.5 and β = 2. Here we take N = 211 . There are two different cases. One corresponds to T = 100 and χ = 20 and the other to T = 150 and χ = 30. The initial data in both cases is u0 (x) = 1, v0 (x) = 2 + random(x), where random(x) is a uniformly distributed in [−0.1, 0.1] random sample. We recover the same chaotic behaviour with merging and emerging peaks. If we track the peaks, we get the results in Figure 7.


38

Figure a

Figure b

40

30

χ=5 χ=20

35 25

30

20

∞

L

||u(t)||

||u(t)||

L

∞

25

20

15

15 10

10

5 5

0

5

10

15

χ

0

20

0

5

10

15

20

25

30

t

Figure 6. Transition to chaos α=1.5,β=2,χ=20

α=1.5,β=2,χ=30

100

150

90

80

70 100

50

t

t

60

40 50 30

20

10

0

−3

−2

−1

0

x

1

2

3

0

−3

−2

−1

0

1

2

3

x

Figure 7. Tracking the peaks for α = 1.5, β = 2 and different values of χ. 12.2.3. α = 1.5, β = 1. Now, we study the case α = 1.5 and β = 1. Here we take N = 213 . We simulate for different values of χ ∈ [16, 20] up to time T = 20. The initial data is given by (36). In this hyperviscous case, the solutions tend to the homogeneous steady state before the instability appears. Then for time 10 < t < 20, kuN (t)kL∞ grows and several peaks emerge (see Figure 8). 12.2.4. α = 0.5, β = 1. We consider the case α = 0.5 and β = 1. Here we take N = 214 . In this hypoviscous case, we simulate the solution corresponding to two different initial data and two different values of the parameter χ (37)

u0 (x) = 1, v0 (x) = 0.1 sin(10x) + 1 and χ = 20,

(38)

u0 (x) = 1, v0 (x) = 0.1 cos(x) exp(−x2 ) + 1 and χ = 10.


39

Figure a

Figure b

9

8

8

t=11.9 t=13.9 t=15.9 t=19.9

7

7 6

6

u(x,t)

||u(t)||L∞

5 5

4

4 3 3

2 2

1

1

0

0

2

4

6

8

10

12

14

16

18

0

20

−3

−2

−1

t

0

1

2

3

x

Figure 8. a) kuN (t)kL∞ and b) uN (ti ) for some 11 < ti < 20 and α = 1.5, β = 1 and χ = 20. α=0.5,β=1,χ=20 Numerical Solution Fitted curve

6000

5000

||∂x u(t)||L∞

4000

3000

2000

1000

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time

Figure 9. k∂x uN (t)kL∞ for some 0 < ti < 0.11 and α = 0.5, β = 1 and χ = 20. The red points are in the fitted curve. In the box, the points that have been used in the fitting process. First, we compute the solution corresponding to (37) with χ = 20 up to time T = 0.11. This solution appears to have a finite time singularity (see Figure 9), i.e. lim sup k∂x u(t)kL∞ = ∞. t→Tmax

Furthermore, if we use least squares to fit a curve with expression a1 (39) y(t) = , (a2 − t)a3 to the numerical solution k∂x uN (t)kL∞ , we obtain the parameters a1 = 0.03389, a2 = 0.11244, a3 = 2.14248. Notice that this evidence of singularity agrees with Theorem 2. On the other


40

α=0.5,β=1,χ=10

α=0.5,β=1,χ=10

4.5

60

t=0 t=0.299 t=0.599 t=0.799

4

3.5

t=0 t=0.299 t=0.599 t=0.799

40

20

∂x u(x,t)

u(x,t)

3

2.5

0

2 −20 1.5 −40 1

0.5

−3

−2

−1

0

1

2

3

−60 −0.25

−0.2

−0.15

−0.1

−0.05

x

0

0.05

0.1

0.15

x

Figure 10. a) u(x, t), b) ∂x uN (t) for some 0 < ti < 0.9 and α = 0.5, β = 1 and χ = 10. hand, if χ is 10, the solution u(t) corresponding to (38) grows in C 1 (see Figure 10) but a curve like (39) does not approximate well the numerical solution uN (t). As a consequence, the value of χ in the formation of a finite time singularity seems to be crucial. Notice that χ is, roughly speaking, 1/r. Consequently, the case r small presents evidence of singularity while the case r big appears to remain smooth for finite time. This is in accordance with [47]. Appendix A. Auxiliary Lemmas We state the Kato-Ponce inequality and the Kenig-Ponce-Vega √ commutator estimate for [Λs , F ]G = Λs (F G) − F Λs G and where Λ = −∆ (see [23, 31, 33]). Lemma 4. Let F, G be two smooth functions on Td . Then we have the following inequalities: k[Λs , F ]GkLp ≤ C(s, p, pi ) (kF kW s,p1 kGkLp2 + kGkW s−1,p3 k∇F kLp4 ) , with 1 1 1 1 1 = + = + , p, p1 , p3 ∈ (1, ∞), p2 , p4 ∈ [0, ∞], s > 0. p p1 p2 p3 p4 and kΛs (F G)kLp ≤

CKP (s, p, pi ) (kΛs F kLp1 kGkLp2 +kΛs GkLp3 kF kLp4 ) , 2

with 1 1 1 1 1 1 = + = + , < p < ∞, 1 < pi ≤ ∞, s > max{0, d/p − d}. p p1 p2 p3 p4 2 Remark 6. In particular, we are using the notation 2α − 2 2 CKP V (α) = C α, 2, 2 + , , ∞, 2 . 2−α α−1 We require the following uniform Gronwall lemma (see [45]).

0.2

0.25


41

Lemma 5. Suppose that g, h, y are non-negative, locally integrable functions on (0, ∞) and dy/dt is locally integrable. If there are positive constants a1 , a2 , a3 , b such that Z t+b Z t+b Z t+b dy y(s)ds ≤ a3 h(s)ds ≤ a2 , g(s)ds ≤ a1 , ≤ gy + h, dt t t t for t ≥ 0, then

y(t + b) ≤

a

3

+ a2 ea1 .

b The last Lemma studies the number of critical points of an analytic function (compare [27]). Lemma 6. Let w > 0, and let u be analytic in the neighbourhood of {z : |ℑz| ≤ w} and 2π-periodic in the x-direction. Then, for any ǫ > 0, T = Iǫ ∪ Rǫ , where Iǫ is an union of at most [ 4π w ] intervals open in T, and • |∂x u(x)| ≤ ǫ, for all x ∈ Iǫ , max|ℑz|≤w |∂x u(z)| . log • card{x ∈ Rǫ : ∂x u(x) = 0} ≤ log2 2 2π w ǫ Appendix B. Explicit expression for the constants

Here we collect explicit expressions for the constants that we use in Theorem 4 and Corollaries 3, 4. We define the radius of the absorbing set in L2 as CKP (α)2 N 3 2 r (40) S(L ) = e 3N + 2CF S (β, α, λ, ν) µ 2α+2 α+1 (CKP (α)CGN (α)) α−1 4N 2 α−1 (2N CF S (β, α, λ, ν)) , + µ and

2 (α) r 2 CSE N 3 (41) I = r + 6N + + 2CF S (β, α, λ, ν) µ ν ν 2 (α) + (C (α))2 1 2CSE I + × 2 µ ! 3 (α)C 1 (α) + C 4 (α)) 2 CKP V (α)(CSE SE SE × , µ The radius of the absorbing set in higher norms is given by 2(1 + e−r ) 2I α/2 2 α/2 2 ˙ e . (42) S(H ) ≤ S(L ) + S(H ) = S(L ) 1 + µ

We denote

2 (43) K1 (α, β, µ, ν, r, λ) = 1+(µ+2+2r+ν+λ+1)CSE (α)+C1 +2C2 +C3 +C4 , 2 (44) K2 (α, β, µ, ν, r, λ) = 1 + (1 + µ + 2 + 2r + ν + λ + 1)CSE (1.1) + C5 + C6 , K1 if α, β > 1, (45) K(α, β, µ, ν, r, λ) = K2 if min{α, β} = 1,

42


where C1 = max 4ωξ − ξ∈R+

(46)

= 4ω

8ω µα

µ α ξ 2

1 α−1

µ − 2

8ω µα

α α−1

,

C2 = max 3ωξ − νξ β ξ∈R+

(47)

= 3ω

3ω νβ

1 β−1

−ν

3ω νβ

β β−1

,

−1/(α−1) (CSI (α))2 α − 1 ω2α (48) C3 = 2 + 2ω ω α (CSI (α))2 + 1.25 + 2r + (49) (50) (51)

C4 = 1.5 +

2 (17.5CSE (α))2

2 (α))2 (9rCSE + , 2

2 2 2 2(CKP V CSE (α)) , µ

2 (1.1))2 (9rCSE µ 2 + 1.25 + 2r + (17.5CSE (1.1))2 + . C5 = 2 4 2 C6 = 1.5 +

2 2 2 2(CKP V CSE (1.1)) . µ

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