Certain methods of proving existence and uniqueness in pde theory

Certain methods of proving existence and uniqueness in pde theory Tomasz Dlotko, Silesian University, Poland Contents 1. Introduction 1.1. Further examples of parabolic problems 2. Existence results 2.1. The method of lines 2.2. Galerkin approximations 2.3. Semigroup approach 2.4. Viscosity technique 3. Uniqueness of solutions 4. References

1 1 1 2 2 8 9 14 17

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1. Introduction In this first lecture I will discuss briefly some basic techniques of proving existence of solutions to pde that are often used in literature. Some known methods of proving uniqueness will be discussed next. To avoid too complicated considerations we will limit our discussion mostly to the case of parabolic equations, which are - in some sense - a direct generalization of the ordinary differential equations to the infinite dimensional case. Physical phenomena like diffusion of substances in liquids, heat transfer in solids, chemical reactions or some biological phenomena will lead to a parabolic equation as a model describing them. The best known example is, of course, the heat equation: ut = ∆u + f (t, x), u(0, x) = u0 (x), t ≥ 0, x ∈ Rn ,

(1)

describing the heat transfer in the space Rn . Another celebrated example is the Navier-Stokes equation describing the fluid flow; R+ 3 t ≥ 0, x ∈ Ω ⊂ R3 , Ω a bounded smooth domain. For the function u = u(t, x), u : R+ × Ω → R3 , f = (f1 , ..., f3 ), we set ut = ν∆u − ∇p − (u, ∇)u + f, divu = 0, t > 0, x ∈ Ω, u = 0 on ∂Ω, u(0, x) = u0 (x).

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1.1. Further examples of parabolic problems. Systems of reaction-diffusion equations. ut = A∆u + f (t, u), + suitable initial - boundary conditions

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Parabolic equations of higher order. ut =

X

(−1)|α| Dα (aαβ Dβ u) + f (t, x),

|α|≤m α

D u(t, x) = 0, |α| ≤ m − 1 on ∂Ω,

(4)

u(0, x) = u0 (x). 2. Existence results There are several methods of proving existence of solutions to parabolic equations. We will discuss here three of them, but let us now call some of the others. In 1950th, for linear problems, this was the method of Fourier series, Fourier transform technique (used for solving the heat equation in Rn ), potential theory i.e. conversion of pde onto

2

integral equation with singular kernel and some approximated methods, like the method of Rothe or the method of lines. Let us describe briefly the last one, based on the monograph [W. Walter]. 2.1. The method of lines. Consider a simple one-dimensional problem: ut = uxx , t ∈ (0, T ], x ∈ (0, a), u = u(t, x), u(0, x) = u0 (x) in [0, a],

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u(t, x) = η0 (t), u(t, a) = η1 (t) for t ∈ (0, T ]. We shall discretize the spatial variable, setting: xν = νh for ν = 0, 1, ..., n, where h = na , and the derivatives will be replaced with the difference quotients; u(t, xν ) by vν (t), ux (t, xν ) by δvν = vν+1h−vν , uxx (t, xν ) by δ 2 vν = h12 (vν + 1 − 2vν + vν+1 ). Then the equation (5) will be transformed onto vν+1 − 2vν + vν−1 vν0 = , ν = 1, 2, ..., n − 1, h2 and the boundary conditions will read vν (0) = u0 (xν ), v0 (t) = η0 (t), vn (t) = η1 (t). The new problem is an initial value problem for a system of (n+1) ode. Solvability of it is simpler than of the original problem; in fact trivial since the system is linear with constant coefficients. Similar method is applicable to a much more general problem: ut = f (t, x, u, ux , uxx ), under suitable monotonicity assumptions on f f (t, x, z, p, r) − f (t, x, z, p, r¯) ≥ α(r − r¯) for r ≥ r¯, |f (t, x, z, p, r) − f (t, x, z, p¯, r)| ≤ L|p − p¯|. For further information see [W. Walther], pp. 275-303. 2.2. Galerkin approximations. In 1969 J.L. Lions has published his monograph devoted to nonlinear pde. For more than 20 years this was the main book used for studying such type problems. A large collection of different methods allowing to study various examples coming from applications has been presented there. We will show how his methods can be applied to a simple example.

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Consider the problem du + Au = f on (0, T ), dt u(0) = u0 ,

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where one could take A = −∆ considered with the Dirichlet boundary condition. Let V, H be two Hilbert spaces, V ⊂ H, V dense in H with continuous inclusion. For our special example we will take H = L2 (Ω) and V = H01 (Ω). The scalar products and the norms are denoted, respectively, by ((·, ·)), (·, ·), k · k, | · |. Identifying H with its dual (space of continuous linear functionals), we have: V ⊂ H ⊂ V 0,

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with continuous inclusions. We will also associate with the continuous linear operator A : V → V 0 , A ∈ L(V, V 0 ) a continuous bilinear form on V × V given by the formula: ∀u,v∈V a(u, v) = (Au, v)V 0 ,V . Thanks to the continuity of A we have that ∀u,v∈V |a(u, v)| ≤ M kukkvk.

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Recall, that a bilinear form will be called coercive, if: ∃α>0 ∀u ∈ V a(u, v) ≥ αkuk2 .

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If a(u, v) = ((u, v)) defines the scalar product in V , then A is a canonical isomorphism of V onto V 0 . We have the well known LaxMilgram lemma: Lemma 1. If a is a bilinear continuous and coercive form on V , then A defines an isomorphism from V into V 0 : ∀A∈L(V,V 0 ) ∃!f ∈V a(x, f ) = A(x) for all x ∈ V. The operator A also defines an isomorphism of its domain D(A) ⊂ H onto H. We will consider problem (6) with u0 ∈ H and f ∈ L2 (0, T ; H) (in particular for f independent on t that means that f ∈ H). Our problem thus reads: ut + Au = f for t ∈ (0, T ), (10) u(0) = u0 , where the equation in (10) is understood in a sense of distributions in V 0 . We are looking for a function u ∈ L2 (0, T ; V ) ∩ C 0 (0, T ; H) such

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that ut = du ∈ L2 (0, T ; V 0 ). To give a sense to the equality in (10) we dt are using the following lemma due to J.L. Lions and R. Temam: Lemma 2. Let X be a Banach space with its dual X 0 and u, g ∈ L1 (a, b; X). The following three conditions are then equivalent: (i) u is a.e. equal to the primitive of g; ∃ξ∈X u(t) = ξ +

Z

t

g(s)ds for a.e. t ∈ [a, b],

0

(ii) for every test function φ ∈ D((a, b)) Z

b

u(t)φ0 (t)dt = −

Z

a 0

b

g(t)φ(t)dt,

a

(iii) for each η ∈ X d (u, η)X 0 ,X = (g, η)X 0 ,X in the sense of scalar distributions on (a,b). dt If one of the above conditions is satisfied, then u is a.e. equal to a continuous function from [a, b] into X. The following existence theorem is formulated in [TE] p. 70. Theorem 1. Under the given assumptions, for u0 ∈ H, f ∈ L2 (0, T : V 0 ), there exists a unique solution u of (10) with u ∈ L2 (0, T : V ) ∩ L∞ (0, T : H), u0 ∈ L2 (0, T : V 0 ). Sketch of the proof. The method of Galerkin approximations will be used in the proof. Assuming that V is separable, consider a base of linearly independent elements in V ; w1 , ..., wm , .... For each fixed m define an approximated solution um of (10): um =

m X

gim wi ,

i=1

d (um , wj ) + a(um , wj ) = (f, wj ), j = 1, ..., m, dt um (0) = u0m ,

(11)

where u0m is a projection in H of u0 onto the linear subspace {w1 , ..., wm }. Often the functions wi will be choosen as orthonormal eigenfunctions of the operator A: ∀j∈N Awj = λj wj . Such a base in H exists provided that (

the form a(u, v) is symmetric i.e. a(u,v) = a(v,u), the inclusion V ⊂ H is compact.

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With this additional assumption the problem (11) becomes simpler and the ode satisfied by the functions gim (t) are separated: m X

um (t) =

gim (t)wi ,

i=1

d gjm (t)(wj , wj ) + λj gjm (t)(wj , wj ) = (f, wj ), j + 1, ..., m, (12) dt gjm (0) = (u0 , wj ). Equation (11) is a linear system of m ode for the unknown functions gim (t) (i=1,...,m). Thus global existence and uniqueness of its solution is obvious. Functions um belongs to C([0, T ] : V ), and derivatives u0m ∈ L2 (0, T : V ). Multiplying equations in (11) by gjm , adding the results for j = 1..., m, we get (u0m , um ) + a(um , um ) = (f, um ), or

1d (um , um ) + ((um , um )) ≤ kf kV 0 kum kV , 2 dt 1d |um |2 + kum k2 ≤ kf kV 0 kum kV . 2 dt Integrating the above over [0, t] and using Cauchy inequality to the right side we get |um (t)|2 + 2

t

Z 0

kum k2 dτ ≤ |um (0)|2 +

Z 0

t

[kum (τ )k2 + kf (τ )k2V 0 ]dτ, (13)

or 2

|um (t)| + 2

Z 0

t

2

2

kum k dτ ≤ |um (0)| +

Z 0

t

kf (τ )k2V 0 dτ.

(14)

Hence the sequence {um } is bounded in L∞ (0, T : H) ∩ L2 (0, T : V ) uniformly in m. The space L2 (0, T : V ) is Hilbert (so; reflexive), L∞ (0, T : H) is a conjugate of L1 (0, T : H), which is separable. Therefore, from a sequence um bounded in such spaces we can extract a subsequence umk , such that umk → u in L2 (0, T : V ) weakly, umk → in L∞ (0, T : V ) weak*. Passing in (11) to the limit we obtain d (u, wj ) + a(u, wj ) = (f, wj ), j + 1, ..., m, dt

(15)

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and, since {wj } forms a basis in V, then also d (u, v) + a(u, v) = (f, v), (16) dt where the equality is understood in the sense of distributions on (0, T ). According to Lemma 1 of [TE], dtd (u, v) = (u0 , v) and we find that the equation (10) is satisfied: ∀v ∈ V

du + Au = f. dt But since f ∈ L2 (0, T : V 0 ) and Au ∈ L2 (0, T : V 0 ), then du ∈ L2 (0, T : dt 0 2 1 V ). Thanks to Lemma 1, and since L ⊂ L , the solution u is a.e. equal to a continuous function from [0, T ] into V 0 . We will also pass to the limit in the initial condition in (11) to see that u(0) = u0 . Remarks. 1. Assuming more regularity of the data we will obtain smoother solution. If f, f 0 ∈ L2 (0, T : H), u0 ∈ D(A), then u0 ∈ C([0, T ] : D(A)), u0 ∈ L2 (0, T : V ) ∩ L∞ (0, T : H). (17) See [TE 2] for the proof. 2. When considering nonlinear problems, additional a priori estimates are required. Having only weak convergence we will not be able to pass to the limit in nonlinear terms. Additional information we need is a uniform in m estimate ku0m kL2 (0,T :V 0 ) ≤ c. Having such an estimate, we already know that: um ∈ L2 (0, T : V ), u0m ∈ L2 (0, T : V 0 ) with the norms bounded uniformly in m. We are thus able to choose a strongly convergent subsequence of um based on the following compactness lemma of J.L. Lions [LI]. Let B0 ⊂ B ⊂ B1 be three Banach spaces, with B0 , B1 reflexive and the inclusion B0 ⊂ B compact. Define W = {v : v ∈ Lp0 (0, T : B0 ), v 0 =

dv ∈ Lp1 (0, T : B1 )} dt

(18)

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where 1 < pi < ∞. W is a Banach space with the norm kvkLp0 (0,T :B0 ) + kv 0 kLp1 (0,T :B1 ) . The announced lemma then holds: Lemma 3. (Compactness lemma) Under the above assumptions, for 1 < pi < ∞, the inclusion W ⊂ Lp0 (0, T : B) is compact. Examples. 1. Consider the Dirichlet problem for the heat equation: ut + ∆u + f (t, x), t ∈ (0, T ), x ∈ Ω ⊂ Rn , u = 0 on ∂Ω, u(0, x) = u0 (x).

(19)

Let V = H01 (Ω), H = L2 (Ω), f ∈ L2 (0, T : H −1 (Ω)), u0 ∈ H. Linear unbounded operator A is given here by D(A) = H 2 (Ω) ∩ H01 (Ω), A is equal to − ∆ with zero Dirichlet condition on D(A) a(u, v) =

n X ∂u ∂v i+1 ∂xi ∂xi

.

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For such data the above presented theory gives existence of a solution u ∈ L2 (0, T : H01 (Ω)) ∩ L∞ (0, T : L2 (Ω)), with u0 ∈ L2 (0, T : H −1 (Ω)). 2. In the case of the fourth order problem: ut = −∆2 u + f (t, x), ∂u = 0 on ∂Ω, u= ∂n u(0, x) = u0 (x),

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we set V = H02 (Ω), H = L2 (Ω), V 0 = H −2 (Ω). Linear operator A equals now (−∆)2 with the above boundary conditions when considered on the domain D(A) = H 4 (Ω) ∩ H02 (Ω). We obtain existence of a solution u ∈ L2 (0, T : H02 (Ω)) ∩ L∞ (0, T : L2 (Ω)),

(22)

with u0 ∈ L2 (0, T : H −2 (Ω)). Remark 1. For the above linear problems uniqueness of solutions is a consequence of the estimates obtained in the proof of existence. Assuming existence of two solutions, we can always consider their difference.

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It will correspond to a solution of the original problem with zero initial and boundary data and zero right hand side. The estimate thus reads: 2

|um (t)| +

Z 0

t

kum (τ )k2 dτ ≤ 0,

giving zero function in H and in V . As a limit, u is also zero function. Probably the largest collection of various methods of proving existence of solutions to pds was given in J.L. Lions monograph [LI]. 2.3. Semigroup approach. Another recent method, because of its great elegance and generality, has growing number of users. Here, in Brasil, it is very popular thanks to the years of activity of Professor Dan Henry. There are three classical references here; the monographs of A. Friedman [FR], A. Pazy [PA] and Dan Henry [HE]. Since the method of Henry is known here I will recall only the main idea and call some generalization. Consider the problem: ut + Au = f (u), t > 0, u(0) = u0 ,

(23)

with sectorial A (see [HE], p.18) (that means −A is an infinitesimal generator of an analytic semigroup). We assume additionally that Re(σ(A)) ≥ a > 0 (if this condition is not satisfied, we need to add the term βI to both sides of (23)). Under such conditions it is possible to define fractional powers of the operator A; Aα , α ∈ R, in particular for α ∈ [0, 1]. Thus A0 = I on X, A1 = A, and we have a family of species X α = D(Aα ), X 0 = X, X 1 = D(A). Definition 1. By a D(Aα )-solution of (23) we mean a continuous function u : [0, τu0 ) → D(Aα ) satisfying (23) in X and such that ut : (0, τu0 ) → X is continuous and u(t) ∈ D(A) for t ∈ (0, τu0 ). As for the case of an ode, having the linear semigroup {e−At } corresponding to linear operator A we can convert this problem into integral equation using the Cauchy formula: u(t) = e−At u0 +

Z

t

e−A(t−s) f (u(s))ds.

(24)

0

Continuous from [0, τu0 ) into X solution of (24) are simultaneously solutions of the differential equation (23). Lipschitz continuity of the nonlinear term of X α into X is sufficient for the local in time solvability of (23), (24). The proof uses Banach

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Fixed Point Theorem. Important in this calculations are the following two estimates of the linear semigroup: s1

ke−At kL(Lp ,Wps1 ) ≤ ct− 2m e−At , t > 0, s−r

ke−At kL(Wpr ,Wps ) ≤ ct− 2m e−At , t > 0,

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where A is defined by a 2m-th order elliptic operator on X = Lp (Ω) (with suitable boundary conditions). Also important is the following property ([HE], pp. 29, 57); if A, f satisfy the above conditions, the resolvent of the operator A is compact and T (t) takes bounded sets into bounded sets for each t > 0, then T (t) is compact on X α , 0 ≤ α < 1, for t > 0. This properties hold for elliptic A on bounded sets Ω. Certain generalization of the above theory was formulated in the monograph [AM] by H. Amann. In this book a generalization of the idea of sectorial operator onto time dependent A(t) is given. We have now more general object than the semigroup; the evolution operator {U(t, s) : 0 ≤ s ≤ t ≤ T } connected with the linear part {A(t), B(t)}. The problem considered there has the form ut + A(t, u)u = f (t, u), t > 0, B(t, u)u = g(t, u), 0 < t ≤ T,

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u(0) = u0 . It will be transformed onto integral equation u(t) = U(t, 0)u0 +

Z

t

U(t, τ )f (τ, u(τ ))dτ.

0

Compare to Henry’s approach the progress is twofold: 1. The main part operator depends on t and u. 2. The boundary conditions are nonlinear. We should mention, that the studies of equations with main part linear operator A(t), having the domain D(A(t)) independent on t belongs to T. Kato, H. Tanabe and P. Sobolevskii (for the discussion of their results see the final part of K. Yosida’s, Functional Analysis). The results of H. Amann as very technical are not suitable for a short presentation here. 2.4. Viscosity technique. We will close our presentation of the techniques of proving existence of solutions to certain pde recalling the so called viscosity technique. There are very deep extensions of this method used, among other, to study complicated equations of fluid dynamics but we will discuss here one of the first application to the classical Korteweg-de Vries equation (see [LI]). This equation has been

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introduced in 1895 by D.J. Korteweg and G. de Vries for description of water waves in long channel. We are looking for a function u = u(t, x), 0 < x < 1, 0 < t < T satisfying the problem: ∂u ∂u ∂3u +u + α 3 = 0, ∂t ∂x ∂x u(0, x) = u0 (x), 0 < x < 1, (27) ∂u ∂u (t, 0) = (t, 1), ..., u(t, 0) = u(t, 1), ∂x ∂x where the boundary conditions above are of periodic type and α is a real constant. For smoother solutions we eventually need to add equalities of higher order derivatives at left and right boundary (x = 0, x = 1). For the considered problem there are infinitely many the so called energy integrals. Multiplying the equation in (27) by u and integrating by parts with the use of boundary conditions we find that d ku(t)kL2 = 0, dt since Z 1 1Z 1 3 2 ux u dx = (u )x dx = 0, 3 0 0 (28) Z 1 3 ∂ u udx = 0, 0 ∂x3 The second energy functional for (27) is obtained through multiplication of the equation by a nonlinear expression: ∂2u . ∂x2 Thanks to the periodic boundary conditions we will get ψ1 (u) = u2 + 2α

Z 0

1

∂u ∂3u u +α 3 ∂x ∂x

!

∂2u u + 2α 2 dx = 0, ∂x !

2

(29)

which leads to the equality Z 0

1

∂u ψ1 (u)dx = 0. ∂t

Thus we get further d Z 1 u3 ∂u − α( )2 dx = 0. dt 0 3 ∂x !

(30)

We are ready to show, using the viscosity method, existence of the solution to (27). We have the following:

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Theorem 2. Let R 3 α 6= 0, and u0 ∈ H 1 (0, 1), u0 (0) = u0 (1). Then there exists a solution u of the problem (27) such that u ∈ L∞ (0, T : H 1 (0, 1)), ut ∈ L∞ (0, T : H −2 (0, 1)). Uniqueness of such a solution is unknown. It will be proved under additional smoothness assumptions on the data. We will use parabolic regularization in the proof of existence. For  > 0 consider the approximating problem ∂u ∂u ∂ 3 u ∂ 4 u + u + α 3 +  4 = 0, (31) ∂t ∂x ∂x ∂x with the same initial and boundary conditions as in the original KdV problem. Using e.g. classical technique of a priori estimates (or the method of compactness, [LI]) it is easy to show existence of solution u to (31) such that (compare Theorem 1 above): u ∈ L∞ (0, T : L2 (0, 1)) ∩ L2 (0, T : H 2 (0, 1)).

(32)

As a result we obtain that ∂u ∈ L2 (0, T : L2 (0, 1)), u ∂x and, from equation (31) it follows that ∂u ∂ 3 u ∂ 4 u ∂u + α 3 +  4 = −u ∈ L2 ((0, T ) × (0, 1)). ∂t ∂x ∂x ∂x Now, linear theory of parabolic equations (compare formula (17)) ensures that: ∂u u ∈ L2 (0, T : H 4 (0, 1)), ∈ L2 ((0, T ) × (0, 1)). (33) ∂t Approximation of solution to (27) by solutions of the problems (31) is called parabolic regularization. To complete this procedure we will need some a priori estimates of solutions u , uniform in . The first estimate is obtained by multiplying (31) by u and integrating next by parts using boundary conditions. It leads to the equality: Z 1 2 1d ∂ u ku k2L2 +  ( 2 )2 dx = 0, 2 dt ∂x 0 and the resulting estimates ku kL∞ (0,T :L2 (0,1)) ≤ c, √ ∂ 2 u k 2 kL2 ((0,T )×(0,1)) ≤ c, ∂x where c denotes a general constant independent on .

(34)

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The second a priori estimate is more involved and similar to the one given earlier for the pure KdV equation. Multiply equation (31) by ψ1 (u ). To continue the considerations we will need the folowing versions of the Nirenberg-Gagliardo type estimates (see e.g. [C-D]): Observation 1. For every v ∈ H 3 ((0, 1)): 11 12 L2 (0,1)

kvkL4 (0,1) ≤ ckvk

d3 v kvkL2 (0,1) + k 3 kL2 (0,1) dx

!

1 12

,

7 ∂v d3 v k kL4 (0,1) ≤ ckvkL122 (0,1) kvkL2 (0,1) + k 3 kL2 (0,1) ∂x dx

!

5 12

.

(35)

Return now to equation (31) multiplied by ψ1 (u ). Dropping the zero components (as previously), we obtain: Z 1 4 ∂u 2 ∂ 2 u ∂ u d Z1 1 3 2 u − α( ) dx +  u + 2α 2 dx = 0, dt 0 3  ∂x ∂x 0 ∂x4 (36) !

!

or, equivalently, Z 1 3 d Z 1 1 3 ∂ u ∂u ∂u 2  u − α( ) dx − 2 u dx dt 0 3 ∂x ∂x 0 ∂x3 !2 Z 1 ∂ 3 u − 2α dx = 0. ∂x3 0

(37)

We thus obtain an estimate: Z 1 ∂u ∂ 3 u ∂ 3 u 2 1d Z1 3 d ∂u 2 u u dx−2 kL2 (0,1) +2αk 3 kL2 (0,1) = dx. α k dt ∂x ∂x 3 dt 0 ∂x ∂x3 0 Integrating the result with respect to t and dividing by α we obtain: Z t ∂ 3 u (s) 2 ∂u 2 du0 2 1 Z1 k kL2 (0,1) + 2 k k ds = k k + u (t, x)3 dx 2 2 ∂x ∂x3 L (0,1) dt L (0,1) 3α 0 0 2 Z t Z 1 ∂u ∂ 3 u 1 Z1 3 (38) u u0 (x)dx dx − − dxds. 3α 0 α 0 0 ∂x ∂x3 Now, thanks to the first estimate in (34) we get: |

Z 0

1

u3 (t)dx| ≤ ku kL∞ (0,1) ku k2L2 (0,1) ≤ c1 ku kL∞ (0,1)

≤ c2

∂u 1+k (t)kL2 (0,1) ∂x

!1 2

≤ c3 +

3|α| ∂u k (t)k2L2 (0,1) . 2 ∂x

(39)

Further, using and the estimates in the Observation, we obtain: Z 1 ∂u ∂ 3 u |α| ∂ 3 u | u dx| ≤ c + k (t)k2L2 (0,1) . (40) 4 ∂x ∂x3 2 ∂x3 0

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Inserting the above estimates into (38) we find that: Z t ∂u 2 ∂ 3 u du0 2 kL2 (0,1) + 2 k 3 (s)k2L2 (0,1) ds ≤ k k 2 ∂x ∂x dx L (0,1) 0 Z t 1 Z1 1 ∂u 2 ∂ 3 u 3 + |u0 | dx + k kL2 (0,1) +  k 3 (s)k2L2 (0,1) ds + c.(41) 3α 0 2 ∂x ∂x 0

k

Reducing similar components we finally obtain k

Z t ∂ 3 u ∂u (t)k2L2 (0,1) + 2 k 3 (s)k2L2 (0,1) ds ≤ c, ∂x ∂x 0

or, that ∂u kL∞ (0,T :L2 (0,1)) ≤ c, ∂x √ ∂ 3 u k 3 kL2 ((0,T )×(0,1)) ≤ c. ∂x k

(42)

Since, thanks to the equation (31), ∂u ∂u ∂ 3 u ∂ 4 u = −u −α 3 − 4 , ∂t ∂x ∂x ∂x it follows from (42) that ∂u is bounded in L2 (0, T : H −2 (0, 1)) as  → 0+ . ∂t We are now ready to pass with  to 0+ in the equation (31). As a consequence of the estimates (34) and (42), we are able to choose a convergent to 0+ sequence {un }, such that ∂u ∂un → in L2 (0, T : L2 (0, 1)) weak*, ∂x ∂x ∂un ∂u → in L2 (0, T : H −2 (0, 1)), ∂t ∂t un → u in L2 ((0, T ) × (0, 1)), and almost everywhere.

un → u,

(43)

in, say, D0 ((0, T ) × (0, 1)), and we are able to Hence un ∂u∂xn → u ∂u ∂x pass in (31) to the limit over such a sequence. Evidently u is a solution of (27). The initial and boundary conditions are also satisfied. The proof is thus completed.

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3. Uniqueness of solutions Classical argument to show uniqueness of solutions to parabolic or elliptic pde is based on the Maximum Principle. We recall here a version of this theorem called Strong Maximum Principle which is suitable for studying linear parabolic equations: Lu =

n X i,j=1

aij (t, x)

n X ∂2u ∂u ∂u + + c(t, x)u − , bi (t, x) ∂xi ∂xj i=1 ∂xi ∂t (44)

where the argument (t, x) vary in an (n + 1)-dimensional set DT = (0, T ] × Ω, and Ω ⊂ Rn is a bounded domain. Assume that: (A) L is parabolic in DT . (B) The coefficients of L are continuous in DT . (C) c(t, x) ≤ 0 in DT . The classical solutions to (44) are assumed to have second continuous x-derivatives and first continuous t-derivative in DT . The Strong Maximum Principle says that Theorem 3. Let the conditions (A), (B), (C) hold. If Lu ≥ 0 in DT and u has in DT a positive maximum which is attained at a point P 0 = (t0 , x0 ) ∈ DT , then u(P ) = u(P 0 ) for all P ∈ S(P 0 ). Here S(P 0 ) = {P = (t, x) ∈ DT : t ≤ t0 }. The proof is given in [FR]. Now the proof of uniqueness of the classical solution of the boundary value problem for (44) is evident. If u1 , u2 are two different solutions of (44) subjected to the same initial-boundary condition: ui (0, x) = u0 (x) for x ∈ Ω, i = 1, 2, ui (t, x) = φ(t, x) for (t, x) ∈ (0, T ] × ∂Ω,

(45)

than their differences u1 − u2 and u2 − u1 must be non-positive in DT (since this functions are equal zero for t = 0 and for x ∈ ∂Ω). For the extension of the above result to nonlinear equations see [WA], p. 199. In Henry’s approach uniqueness of solutions is guaranteed by the Lipschitz continuity on bounded subsets of X α of the nonlinear term f . Uniqueness is proven directly inside the proof of the local solvability (based on the contracting mapping principle). It will be also shown independently. Let u1 , u2 be two local solutions of the integral equation (24) −At

u(t) = e

u0 +

Z 0

t

e−A(t−s) f (u(s))ds,

15

considered on their common interval of existence. Both the solutions correspond to the same initial function u0 , so that t

Z

u1 (t) − u2 (t) =

0

e−A(t−s) [f (u1 (s)) − f (u2 (s))]ds,

(46)

for small t > 0. Applying X α norm to both sides of (46) we get: ku1 (t) − u2 (t)kxα ≤ ≤

t

Z

kf (u1 (s)) − f (u2 (s))kX ds

0 t

Z

(t − s)−α e−a(t−s) Lku1 (s) − u2 (s)kX α ds. (47)

0

The last integral inequality will be studied with the use of Lemma 7.1.1, p. 188 in[HE]. Thanks to evident estimate (t − s)−α e−a(t−s) ≤ (t − s)−α , s ≤ t, for v(t) := ku1 (t) − u2 (t)kX α an estimate v(t) ≤ c

0

Z

t

(t − s)−α v(s)ds

0

is satisfied by the trivial function v ≡ 0. For completness of the presentation we call here the corresponding lemma from [HE]. Lemma 4. Let β > 0, b ≥ 0 and let a be a nonnegative and locally integrable function on 0 ≤ t < τ . Let a nonnegative and locally integrable function u satisfies for 0 ≤ t < τ u(t) ≤ a(t) + b

Z

t

(t − s)β−1 u(s)ds.

(48)

0

Then, u(t) ≤ a(t) + θ

t

Z 0

nβ

Eβ0 (θ(t − s))a(s)ds, 0 ≤ t < τ, 1

z 0 β where Eβ (z) = ∞ n=0 Γ(nβ+1) , θ = [bΓ(β)] , Eβ (z) = ticular, for a(t) ≡ a = const.,

P

d E (z). dz β

In par-

u(t) ≤ aEβ (θt). The above lemma extends the classical Gronwall Lemma. Strange examples Usually uniqueness of solutions is a natural property of solutions to parabolic problems. However, even for solutions of the familiar heat equation in Rn one should be carefull. An unexpected example below belongs to Tychonov (1935) (see [FR 2], p. 30).

16

Example. For dimension n = 1 and arbitrary δ > 0 there exists a C ∞ -smooth function f : R → R, not identically equal zero; f (t) = 0, outside [0, 1], satisfying |f (m) (t)| ≤ C m m(1+δ)m , m = 1, 2, 3, .... for t ∈ [0, 1]. Define u(t, x) =

∞ X f (m) (t) m=0

(2m)!

x2m .

When δ < 1, the series together with its two first derivatives is uniformly convergent in bounded sets. Also, for some  > 0 the condition Z 0

1

Z

∞

|u(t, x)|exp(−k|x|2+ )dxdt ≤ ∞,

(49)

−∞

is satisfied. It is easy to see that the Cauchy problem: ut = uxx , t > 0, x ∈ R, u(0, x) = 0, x ∈ R,

(50)

has two different solutions; the one constructed above and zero function. Example. For the celebrated Navier-Stokes equation in dimension 3 existence of a global in time weak solution u in the sense of Lions is shown such that 8

u ∈ L 3 (0, T : L4 (Ω)), 4

ut ∈ L 3 (0, T : V 0 ).

(51)

Here, V = clH01 (Ω) D, where D = {φ ∈ D(Ω) : divφ = 0} (D is the space of test functions C0∞ (Ω). The smoothness properties of such solution are not sufficient to prove its uniqueness (the nonlinear term in the Navier-Stokes equation need not be locally Lipschitz under such assumptions). To have uniqueness we need additional smoothness: u ∈ L∞ (0, T : H) ∩ L∞ (0, T : V ), ut ∈ L∞ (0, T : L4 (Ω)).

(52)

17

4. References [AM] H. Amann, Linear and Quasilinear Parabolic Problems, Birkh¨auser, Basel, 1995. [C-D] J. W. Cholewa, T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000. [FR] A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969. [FR 1] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, INC., Englewood Cliffs, N.Y., 1964. [FU] H. Fujita, On the blowing up of solutions of the Cauchy problem for ut = ∆ + u1+α , J. Fac. Sci. Univ. Tokyo Sect. A Math. 16 (1966), 109-124. [HA] J. K. Hale, Asymptotic Behavior of Dissipative Systems, AMS, Providence, RI, 1988. [HE] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, 1981. [KR] S. G. Kre˘ın, Linear Equations in Banach Spaces. Translated from the Russian (1971) by A. Iacob. Birkhauser, Boston, Mass., 1982. [LA] O.A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. [LI] J.L. Lions, Quelques M´ethodes de R´esolution des Problemes aux Limit`es Non Lin´eaires, Dunod, Paris, 1969. [LU] A. Lunardi, Analytic Semigroup and Optimal Regularity in Parabolic Problems, Birkh¨auser, Berlin, 1995. [PA] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [TE] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [RT 2] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3ed Edition, North-Holland, Amsterdam, 1984. [WA] W. Walter, Differential and Integral Inequalities, Springer-Verlag, New York, 1970.