Local Existence and Blow-up Criterion for the Euler Equations in the Besov Spaces Dongho Chae∗ Department of Mathematics Seoul National University Seoul 151-742, Korea e-mail :
[email protected] [published in Asymptotic Analysis, 38, no. 3-4, (2004),339-358]
Abstract We prove the local in time existence and a blow-up criterion of solutions in the Besov spaces for the Euler equations of inviscid incompressible fluid flows in Rn , n ≥ 2. As a corollary we obtain the persistence of Besov space regularity for the solutions of the 2-D Euler equations with initial velocity belonging to the Besov spaces. For the proof of the results we establish a logarithmic inequality of the Beale-Kato-Majda type and a Moser type of inequality in the Besov spaces.
1
Introduction and Main Results
We are concerned by the Euler equations for homogeneous incompressible fluid flows. ∂v + (v · ∇)v = −∇p, (x, t) ∈ Rn × (0, ∞), (1.1) ∂t div v = 0, (x, t) ∈ Rn × (0, ∞), (1.2) ∗
This research is supported partially by the grant no.2000-2-10200-002-5 from the basic research program of the KOSEF.
1
x ∈ Rn ,
v(x, 0) = v0 (x),
(1.3)
where v = (v1 , · · · , vn ), vj = vj (x, t), j = 1, · · · , n, is the velocity of the fluid, p = p(x, t) is the scalar pressure, and v0 is the given initial velocity satisfying div v0 = 0. Given v0 ∈ H m (Rn ), for integer m > n2 + 1, Kato proved the local in time existence and uniqueness of solution in the class C([0, T ]; H m (Rn )), where T = T (kv0 kH m )([13]). Temam extended this result to bounded domains, in H m (Ω) and in W m,p (Ω)(See Ref. [22, 23]). Later, Kato and Ponce obtained a local existence result in the more general Sobolev s space, W s,p (Rn ) = (1 − ∆)− 2 Lp (Rn ) for real number s > n/p + 1([14]). One of the most outstanding open problems in the mathematical fluid mechanics is to prove the global in time continuation of the local solution, or to find an initial data v0 ∈ H m (Rn ) such that the associated local solution blows up in finite time for n = 3. One of the most significant achievements in this direction is the Beale-Kato-Majda criterion[3] for the blow-up of solutions, which states that lim sup kv(t)kH m = ∞ (1.4) t%T∗
if and only if
Z
T∗
kω(s)kL∞ ds = ∞,
(1.5)
0
where ω =curl v is the vorticity of the flow. Recently this criterion has been refined by Kozono and Taniuch[16], replacing the L∞ norm of the vorticity by the BMO norm.(We note L∞ ,→ BM O. See e.g. [21] for a detailed description of the space BMO.) In the H¨older space, Lichtenstein[17], Chemin[10] proved the local existence of solution. Bahouri and Dehman[2] also obtained the blow-up criterion for the local solution in the H¨older space. We also mention that there is a geometric type of blow-up criterion, using the deep structure of the nonlinear term of the Euler equation[11]. For n = 2 it is wellknown that the local solution can be continued for all time, and one of the main questions in this case is persistence problem of the regularity of initial data. Kato and Ponce proved the persistence of the Sobolev space regularity for the super critical Sobolev space initial data, i.e v0 ∈ W s,p (R2 ) with s > 2/p+1[15]. All the above results are concerened only in the super-critical cases, namely s > n/p + 1. In the current paper we study the initial value problem for the initial data belonging to the both super-critical and critical s , s > n/p + 1 with p ∈ (1, ∞), q ∈ [1, ∞], Besov spaces, namely v0 ∈ Bp,q or s = n/p + 1, q = 1, and obtain the blow-up criterion in the Besov space as well as the local in time existence of solutions. Our criterion is sharper than the Beale-Kato-Majda’s and the Kozono-Taniuch’s result, in the sense 0 norm, which is that the BMO norm of vorticity is replaced by the B˙ ∞,∞ 0 ˙ weaker than the BMO norm (remember BM O ,→ B∞,∞ ). We also recall the s = H s (Rn ) between the Besov space and the fractional order relation, B2,2 Sobolev space. As a corollary of our criterion we prove the global in time 2
existence and persistence of Besov space regularity in the 2-dimensional Eus ler equations for v0 ∈ Bp,q , s > 2/p + 1. We mention that Vishik obtained the global well-posedness of the 2-dimensional Euler equations in the critical Besov space([25, 26]). We also mention that in the case of the Navier-Stokes equations in the Besov spaces there are many studies by Cannone and his collaborators(See Ref.[6, 7] and references therein.). In order to prove our blow-up criterion we establish the following logarithmic Besov space inequality with features similar to the logarithmic Sobolev inequality in [3]. Proposition 1.1 Let s > n/p with p ∈ (1, ∞), q ∈ [1, ∞]. There exists a s the following inequality holds: constant C such that for all f ∈ Bp,q s + 1)) (log+ kf kBp,q kf kL∞ ≤ C(1 + kf kB˙ ∞,∞ 0
(1.6)
The following is our main theorem. Theorem 1.1 (Main Theorem) (i) Local in time existence: Let s > n/p + 1 with p ∈ (1, ∞), q ∈ [1, ∞], or s s = n/p + 1 with p ∈ (1, ∞), q = 1. Suppose v0 ∈ Bp,q , satisfying div s ) such that a unique v0 = 0, is given. Then, their exists T = T (kv0 kBp,q s solution v ∈ C([0, T ]; Bp,q ) of the system (1.1)-(1.3) exists. (ii) Blow-up criterion: A. Super-critical case: Let s > n/p + 1, p ∈ (1, ∞), q ∈ [1, ∞]. Then, s the local in time solution v ∈ C([0, T ]; Bp,q ) blows up at T∗ > T in s Bp,q , namely s lim sup kv(t)kBp,q =∞ (1.7) t%T∗
if and only if
Z
T∗
kω(t)kB˙ ∞,∞ dt = ∞. 0
0
(1.8)
B. Critical case: Let p ∈ (1, ∞). Then, the local in time solution n/p+1 n/p+1 v ∈ C([0, T ]; Bp,1 ) blows up at T∗ > T in Bp,1 , namely lim sup kv(t)kB n/p+1 = ∞ p,1
t%T∗
if and only if
Z
(1.9)
T∗ 0
kω(t)kB˙ ∞,1 dt = ∞. 0
(1.10)
0 s , the above = H s (Rn ), and L∞ ,→ BM O ,→ B˙ ∞,∞ Remark 1.2 Since B2,2 theorem improves the original Beale-Kato-Majda criterion[3], and its refined version by Kozono and Taniuchi[16] for the case of H s (Rn ). On the other
3
s hand, since B∞,∞ = C 0,s (Rn ), we find that the result of blow-up criterion in the H¨older space by Bahouri and Dehman[2] corresponds to the extreme case of Theorem 1.1 (ii),A.
If n = 2 the conservation of vorticity, kω(t)kL∞ = kω0 kL∞ for all t > 0, s−1 0 combined with the embedding Bp,q ,→ L∞ (R2 ) ,→ B˙ ∞,∞ for s > 1 (See Remark 2.1 below.), implies immediately the following corollary. Corollary 1.1 (Persistence of the Besov space regularity) Let s > 2/p+ s 1 with p ∈ (1, ∞), q ∈ [1, ∞]. Suppose v0 ∈ Bp,q , satisfying div v0 = 0, is s ) to the system given. Then there exists a unique solution v ∈ C([0, ∞) : Bp,q (1.1)-(1.3) with n = 2. s Remark 1.3 In the case p = q = 2, since B2,2 = H s (R2 ), the Lebesgue space, we recover the result by Kato-Ponce[15].
The results in this paper were announced in [8].
2
Proof of the Main Results
We first set our notations, and recall definitions on the Besov spaces. We follow [20] and [24]. Let S be the Schwartz class of rapidly decreasing functions. Given f ∈ S its Fourier transform F(f ) = fˆ is defined by Z 1 fˆ(ξ) = e−ix·ξ f (x)dx. n/2 (2π) Rn We consider ϕ ∈ S satisfying Supp ϕˆ ⊂ {ξ ∈ Rn | 12 ≤ |ξ| ≤ 2}, and ϕ(ξ) ˆ > 0 if 23 < |ξ| < 32 . Setting ϕˆj = ϕ(2 ˆ −j ξ) (In other words, ϕj (x) = jn j 2 ϕ(2 x).), we can adjust the normalization constant in front of ϕˆ so that (See e.g. Lemma 6.1.7,[4]) X ϕˆj (ξ) = 1 ∀ξ ∈ Rn \ {0}. j∈Z
Given k ∈ Z, we define the function Sk ∈ S by its Fourier transform X Sˆk (ξ) = 1 − ϕˆj (ξ). j≥k+1
ˆ In particular we set Sˆ−1 (ξ) = Φ(ξ). We observe Supp ϕˆj ∩ Supp ϕˆj 0 = ∅ if |j − j 0 | ≥ 2.
4
(2.11)
Let s ∈ R, p, q ∈ [0, ∞]. Given f ∈ S 0 , we denote ∆j f = ϕj ∗ f , and then the homogeneous Besov norm kf kBp,q s ˙ is defined by kf kB˙ p,q = s
( £ P∞
2jqs kϕj ∗ f kqLp supj [2js kϕj ∗ f kLp ] −∞
¤ 1q
if q ∈ [1, ∞) . if q = ∞
s The homogeneous Besov space B˙ p,q is a semi-normed space with the seminorm given by k · kB˙ p,q s . For s > 0, p, q ∈ [0, ∞] we define the inhomogeneous s of f ∈ S 0 as Besov space norm kf kBp,q s kf kBp,q = kf kLp + kf kB˙ p,q s .
The inhomogeneous Besov space is a Banach space equipped with the norm, s . (See e.g. [4, 20, 24]). We now recall the following lemma[10]. k · kBp,q Lemma 2.1 (Bernstein’s Lemma) Assume that f ∈ Lp , 1 ≤ p ≤ ∞, and Supp fˆ ⊂ {2j−2 ≤ |ξ| < 2j }, then there exists a constant Ck such that the following inequality holds Ck−1 2jk kf kLp ≤ kDk f kLp ≤ Ck 2jk kf kLp .
(2.12)
As an immediate corollary of the above lemma we have the equivalence of norms, (2.13) kDk f kB˙ p,q ∼ kf kB˙ p,q s+k . s We are now ready to prove Proposition 1.1. Proof of Proposition 1.1: We decompose X X ∆j f (x) + ∆j f (x) f (x) = S−N ∗ f (x) + j≥N
|j|
