Université de Metz. Master 2 Recherche de Mathématiques 2ème semestre. par Ralph Chill Laboratoire de Mathématiques et Applications de Metz

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1 Université de Metz Master 2 Recherche de Mathématiques 2ème semestre Systèmes gradients par Ralph Chill Laboratoire de Mathématiques et Applications de Metz Année 26/7 1

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3 Contents Chapter 1. Introduction 5 Chapter 2. Linear gradient systems 7 1. Definition of gradient systems 7 2. Operators associated with bilinear forms 9 3. The theorem of J.-L. Lions * L p -maximal regularity * Interpolation and L p -maximal regularity 22 Chapter 3. Nonlinear gradient systems Quasilinear equations: existence and uniqueness of local solutions Regularity of solutions * Navier-Stokes equations: local existence of regular solutions * Diffusion equations: comparison principle * Energy methods and stability 53 Chapter 4. Appendix Closed linear operators Vector-valued L p spaces Vector-valued Sobolev spaces 61 Bibliographie 63 3

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5 CHAPTER 1 Introduction A gradient system in finite dimension is an ordinary differential equation of the form (.1) u + ϕ(u) =, where ϕ C 1 (R n ; R) is a function and ϕ is its euclidean gradient: ϕ = ( 1 ϕ,..., n ϕ). Every solution u of the gradient system (.1) has the important property that ϕ is decreasing along u, that is, the function ϕ(u) is decreasing. This follows simply from derivating: d dt ϕ(u(t)) = ϕ(u(t)), u(t) = u(t) 2. In this equation,, denotes the euclidean scalar product and the corresponding euclidean norm. In fact, the equation (.1) just says that the time derivative of u is opposite to the gradient ϕ(u) which shows into the direction into which the function ϕ has largest directional derivative with respect to a unit vector in the euclidean norm. A solution of the gradient system (.1) thus always tries to minimize the value ϕ(u) as fast as possible in the given geometry (here, the euclidean geometry). The gradient system (.1) admits ϕ as Lyapunov function or energy function: by this we just mean the fact that ϕ(u) is decreasing along every solution. Having a Lyapunov/energy function is very natural in examples of ordinary differential equations arising in physics when the quantity ϕ(u) has an interpretation of a real energy. In the following, we will often call ϕ an energy function. In this course we will study gradient systems in finite and infinite dimension, with an emphasis on the infinite dimensional case. Examples of infinite dimensional gradient systems include the linear heat equation and the semilinear heat equation u t u = f u t u + f (u) = 5

6 6 1. INTRODUCTION on an open subset of R n. These parabolic partial differential equations can in fact be rewritten as ordinary differential equations on infinite dimensional Hilbert spaces spaces. The resulting ordinary differential equations are gradient systems. Other examples of gradient systems are given by certain geometric evolution equations like the mean curvature flow (curve shortening flow in one dimension), the surface diffusion flow, the Willmore flow, and the Ricci flow. The latter has recently played an important role in the (very probable) solution of the Poincaré conjecture, one of seven millenium problems. An introduction to at least one of these flows will be aim of this course. In a first place, however, we will have to provide some basic material necessary for studying gradient systems in infinite dimensions. After these preliminaries, we will study existence and uniqueness of solutions of linear gradient systems. The next step will bring us to prove existence and uniqueness of solutions of certain nonlinear evolution equations. Eventually, we will also study their regularity properties. While we will always test our abstract results in concrete examples, we will only at the end of this course be able to turn to geometric evolution equations.

7 CHAPTER 2 Linear gradient systems Throughout we denote by X and Y (real) Banach spaces and by H, K, V (real) Hilbert spaces. The norm on a Banach space X is usually denoted by X or, and the inner product on a Hilbert space H is usually denoted by (, ) H or (, ). Recall that a linear operator T : X Y is continuous if and only if it is bounded, i.e. if and only if T L(X,Y) := sup x X 1 T x Y is finite. Instead of speaking of continuous linear operators we will in the following speak of bounded linear operators. The space of all bounded linear operators from X into Y is denoted by L(X, Y). It is a Banach space for the norm L(X,Y). 1. Definition of gradient systems Let V be a real Hilbert space with inner product, V and let ϕ : V R be a function of class C 1. At every point u V the derivative ϕ (u) is by definition an element of V and therefore ϕ is a function V V. Let H be a second real Hilbert space with inner product, H and suppose that V is a dense subspace of H and that the embedding of V into H is bounded, i.e. there exists a constant C such that u H C u V for every u V. We will write V H for this situation. In the following, we denote by V the dual space of V, i.e. V = L(V, R). The duality between V and V is denoted by the bracket, V,V. As soon as V is densely and continuously embedded into H, the dual H is densely and continuously embedded into the dual V. In fact, the restriction to V of a bounded linear functional u H defines a bounded linear functional in V. The resulting operator H V is clearly linear and bounded, and it is injective by the fact that V is dense in H. Using reflexivity of Hilbert spaces, one can even show that the embedding of H into V is dense. Hence, if V H, then H V. We recall the theorem of Riesz-Fréchet which says that for every bounded linear functional u H there exists a unique element u H such that u, v H,H = u, v H for every v H. On the other hand, it is clear from the bilinearity of the inner product that for every u H the functional u : v u, v H is linear and bounded, i.e. it belongs to H. So 7

8 8 2. LINEAR GRADIENT SYSTEMS the theorem of Riesz-Fréchet allows us to identify the spaces H and H via the linear isomorphism u u. This isomorphism is even isometric as one easily verifies. It will be a convention in the following that we will always identify H and H via this isomorphism. We write H = H, but we have in mind that this equality does not hold in the set theoretic sense and that the isomorphism behind this equality depends on the choice of the inner product in H. By our assumption that V H and by our convention that H = H, we thus obtain the following picture (1.1) V H = H V, and in particular V is densely and continuously embedded into V (but we can not identify V and V once we have identified H and H! The space V is only a subspace of V ). The above chain implies that for every u V and every v H (1.2) u, v V,V = u, v H = u, v H,H. By a gradient system we will understand an evolution equation of the form (1.3) u + ϕ (u) =. Classical solutions of this gradient system will be continuously differentiable functions u : [, T] V for which the equality (1.3) holds in the space V : recall that ϕ (u) is an element of V, that u is an element of V and that V is a subspace of V by our convention. We emphasize the fact that this evolution equation depends on the choice of the Hilbert space H and in particular on the choice of the inner product in H. Sometimes, it will therefore be convenient to write H ϕ(u) instead of the derivative ϕ (u). If the Hilbert space H is clear from the context, it suffices to write ϕ(u). EXAMPLE 1.1. Let V = R n and H = R n equipped with the euclidean inner product. Let ϕ : R n R be of class C 1. By Riesz-Fréchet, for every u R n there exists R nϕ(u) such that ϕ (u), v (R n ),R n = R nϕ(u), v R n for every v Rn. It is easy to verify that R nϕ = ϕ is the euclidean gradient of ϕ, i.e. R nϕ(u) = ( 1 ϕ(u),..., n ϕ(u)). The resulting gradient system is the system (.1) from the Introduction. EXAMPLE 1.2. We let again V = R n and H = R n but we equip H with the inner product u, v H := Qu, v R n, where Q is a symmetric and positive definite matrix. By Riesz-Fréchet, for every u R n there exists H ϕ(u) such that ϕ (u), v (R n ),R n = Hϕ(u), v H for every v R n.

9 2. OPERATORS ASSOCIATED WITH BILINEAR FORMS 9 On the other hand, by the definition of the scalar product in H and by the previous example H ϕ(u), v H = Q H ϕ(u), v R n = R nϕ(u), v R n. Since this equality holds for every v R n, we obtain H ϕ(u) = Q 1 R nϕ(u). The resulting gradient system is u + Q 1 ϕ(u) =, where ϕ denotes the euclidean gradient. 2. Operators associated with bilinear forms In this section, V will be a real Hilbert space with inner product, V. DEFINITION 2.1 (Bilinear form). A function a : V V R is called a bilinear form if it is linear in each variable, i.e. a(αu + βv, w) = αa(u, w) + βa(v, w) and a(u, αv + βw) = αa(u, v) + βa(u, w) for every u, v, w V and every α, β K. There are some simple but important examples of bilinear forms. EXAMPLE 2.2. Every inner product on V is a bilinear form! EXAMPLE 2.3. Let V = H 1 () ( Rn open) be the Sobolev space which is obtained by taking the closure of D() (the test functions on ) in H 1 (). The space V is equipped with the inner product (u, v) H 1 := uv + u v, and the corresponding norm On this Sobolev space the equality a(u, v) := defines a bilinear form. u H 1 = ( u 2 L 2 + u 2 L 2 ) 1 2. u v, u, v V, EXAMPLE 2.4. More generally, if A L (; R n n ) is a bounded, measurable, matrix valued function, then the equality a(u, v) := A(x) u v, u, v V, defines a bilinear form on the Sobolev space V = H 1 (). DEFINITION 2.5 (Boundedness, coercivity, symmetry). Let a be a bilinear form on a Hilbert space V.

10 1 2. LINEAR GRADIENT SYSTEMS (a) We say that a is bounded if there exists a constant C such that a(u, v) C u V v V for every u, v V. (b) We say that a is coercive if there exists a constant η > such that (c) We say that a is symmetric if Re a(u, u) η u 2 V for every u V. a(u, v) = a(v, u) for every u, v V. DEFINITION 2.6 (Operator associated with a bounded bilinear form). Given a bounded, bilinear form a on V, we define the linear operator A : V V associated with this form by Au, ϕ V,V := a(u, ϕ), u, ϕ V. It follows from the boundedness of a that the operator A is well-defined and bounded. In fact, let C be the constant from Definition 2.5 (a). Then Au V = sup Au, ϕ V,V ϕ V 1 = sup ϕ V 1 a(u, ϕ) sup C u V ϕ V = C u V. ϕ V 1 The following theorem says something about the solvability of the equation Au = f for given f V. As one can see from the statement, coercivity of a implies invertibility of A. THEOREM 2.7 (Lax-Milgram). Let a be a bounded, coercive, bilinear form on V. Then for every f V there exists a unique u V such that a(u, ϕ) = f, ϕ V,V for every ϕ V. PROOF. We have to prove that the bounded linear operator A L(V, V ) associated with a is bijective. By coercivity, for every u V \ {}, Au V = sup Au, v V,V v V 1 u Au, V u,v V = 1 a(u, u) u V η u V. This proves on the one hand injectivity of A, but also that Rg A is closed in V. If Rg A V, then there exists v V \ {} such that Au, v V,V = for every u V. If we take u = v, then we obtain = Av, v V,V = a(v, v) η v 2 V >, a contradiction. Hence, Rg A = V, i.e. A is surjective.

11 2. OPERATORS ASSOCIATED WITH BILINEAR FORMS 11 DEFINITION 2.8. Let V and H be two Hilbert spaces such that (1.1) holds. We call a form a : V V R H-elliptic if there exists ω R such that the form a ω : V V R defined by a ω (u, v) := a(u, v) + ω(u, v) H is coercive, i.e. if there exists η > such that a(u, u) + ω u 2 H η u 2 V for every u V. DEFINITION 2.9. We call a matrix A R n n elliptic if there exists a constant η > such that Aξ ξ η ξ 2 for every ξ C n. We call a matrix-valued function A L (, R n n ) uniformly elliptic if there exists a constant η > such that Re A(x)ξ ξ η ξ 2 for every ξ C n, x. In the above definition, if the matrix A is symmetric then ellipticity of A is equivalent to saying that A is positive definite. EXAMPLE 2.1. Take the bilinear form a from Example 2.4 and assume that A L (, R n n ) is uniformly elliptic. Then a is bounded and elliptic. Indeed, a(u, u) + η u 2 = A(x) u u + η u 2 L 2 L 2 η u 2 + η u 2 L 2 = η u 2 H 1. We define a second operator associated with a form a. DEFINITION 2.11 (Operator associated with a bilinear form). Let a be a bounded, bilinear form on V, and let H be a second Hilbert space such that (1.1) holds. We define the operator A H : H D(A H ) H associated with a by D(A H ) := {u V : v H ϕ V : a(u, ϕ) = (v, ϕ) H }, A H u = v. The operator A H is well-defined in the sense that the element v H is uniquely determined if it exists. Indeed, assume that there are two elements v 1, v 2 H such that (v 1, ϕ) H = a(u, ϕ) = (v 2, ϕ) H for every ϕ V. Then (v 1 v 2, ϕ) H = for every ϕ V, and since V is dense in H (here the density of the embedding is used!), this already implies v 1 = v 2. LEMMA The operator A H is the restriction of A to the space H, i.e. PROOF. Exercise. D(A H ) = {u V : Au H} and A H u = Au for u D(A H ).

12 12 2. LINEAR GRADIENT SYSTEMS EXAMPLE 2.13 (Dirichlet-Laplace operator). Let V = H 1 (, 1), H = L2 (, 1) and consider the form a : H 1 H1 R defined by a(u, v) = 1 u v. The dual space of H 1 is denoted by H 1. Let A : H 1 H 1 be the operator associated with the form a and let A L 2 be its restriction to L 2. We show that D(A L 2) = H 2 (, 1) H 1 (, 1) and A L 2u = u. This operator is called the Dirichlet-Laplace operator on the interval (, 1). Let u D(A L 2) and let f = Au L 2. Then, for every ϕ H 1 one has f, ϕ H = Au, ϕ H = Au, ϕ H 1,H 1 = a(u, ϕ), or (2.1) 1 f ϕ = 1 u ϕ for every ϕ H 1 (, 1) By the definition of H 1, this means that u H 1 (, 1) and u := (u ) = f. In other words, u H 2 (, 1) and Au = u. On the other hand, let u H 2 H 1 and let f = u L 2. One easily shows that (2.1) holds, so that f, ϕ H = a(u, ϕ) for every ϕ H 1. By definition, this implies u D(A L 2) and Au = u. 3. The theorem of J.-L. Lions Throughout this section, V and H are two real separable Hilbert spaces such that V H = H V, with dense injections. Moreover, we let a : V V R be a bounded bilinear form, and we let A : V V be the operator associated with A. Then we consider the evolution equation (3.1) u(t) + Au(t) = f (t), t [, T], u() = u. This evolution equation is a gradient system if the form a is in addition symmetric. The underlying energy is then ϕ : V R, ϕ(u) = 1 a(u, u). In fact, the 2 derivative of this quadratic form can be calculated very easily using the definition of

13 3. THE THEOREM OF J.-L. LIONS 13 the Fréchet derivative and the bilinearity and the symmetry of the form a: i.e. ϕ (u) = Au. ϕ(u + h) = 1 ( ) a(u, u) + a(u, h) + a(h, u) + a(h, h) 2 = ϕ(u) + a(u, h) + 1 a(h, h) 2 = ϕ(u) + ϕ (u), h V,V + o(h), THEOREM 3.1 (J.-L. Lions). Let a : V V R be a bilinear, bounded, elliptic form and let A : V V be the associated operator. Let T >. Then for every f L 2 (, T; V ) and every u H there exists a unique solution u W 1,2 (, T; V ) L 2 (, T; V) of the problem (3.1). We will prove this theorem in several steps. First, we study the maximal regularity space MR 2 (a, b; V, V). LEMMA 3.2. For every u W 1,2 (R; V ) L 2 (R; V) the function t 1 2 u(t) 2 H is differentiable almost everywhere and (3.2) 1 d 2 dt u(t) 2 H = u(t), u(t) V,V. PROOF. One shows by regularisation that the space C 1 c(r; V) is dense in W 1,2 (R; V ) L 2 (R; V). Then one verifies that for functions u C 1 c(r; V) the equality (3.2) is true, using also the equality d 1 dt 2 u(t) 2 H = ( u(t), u(t)) H = u(t), u(t) V,V. The claim then follows by an approximation argument. LEMMA 3.3. One has (3.3) W 1,2 (R; V ) L 2 (R; V) C (R; H). PROOF. We use again the fact that the space Cc(R; 1 V) is dense in W 1,2 (R; V ) L 2 (R; V). For every u Cc(R; 1 V) and every t R one has u(t) 2 H = = 2 = 2 t t t d ds u(s) 2 H ds ( u(s), u(s)) H ds u(s), u(s) V,V ds 2 u L 2 (R;V ) u L 2 (R;V) u 2 L 2 (R;V ) + u 2 L 2 (R;V) 2 u 2 MR 2 (R;V,V).

14 14 2. LINEAR GRADIENT SYSTEMS Hence, the embedding operator (C 1 c(r; V), MR2 (R;V,V)) (C (R; H), C (R;H)) is bounded. Since Cc(R; 1 V) is dense in W 1,2 (R; V ) L 2 (R; V), the embedding (3.3) follows. LEMMA 3.4. For every < a < b < one has (3.4) W 1,2 (a, b; V ) L 2 (a, b; V) C([a, b]; H). PROOF. Let a, b be arbitrary, but finite. There exists a linear bounded extension operator E : W 1,2 (a, b; V ) L 2 (a, b; V) W 1,2 (R; V ) L 2 (R; V) with the property that Eu restricted to the interval (a, b) equals u (exercice!). Using that the restriction operator C (R; H) C([a, b]; H), u u [a,b] is linear and bounded, too, the claim follows by considering the composition of the extension operator E, the embedding (3.3), and this restriction operator. LEMMA 3.5 (Uniqueness). Let A be as in Theorem 3.1. Then for every f L 2 (, T; V ) and every u H there exists at most one solution u W 1,2 (, T; V ) L 2 (, T; V) of the problem (3.1). PROOF. By linearity, it suffices to prove that if u W 1,2 (, T; V ) L 2 (, T; V) is a solution of u(t) + Au(t) =, t [, T], u() =, then u =. So let u be a solution of this problem. Then, by ellipticity of the form a, and by Lemma 3.2, As a consequence, Hence, u =. 1 d 2 dt u(t) 2 H = u(t), u(t) V,V = Au(t), u(t) V,V ω u(t) 2 H. u(t) 2 H e 2ωt u() 2 H = for every t [, T]. PROOF OF THEOREM 3.1. By Lemma 3.5 it remains only to prove existence of a solution. The proof of existence will be done by a Galerkin approximation. Let (w n ) V be a linearly independent sequence such that span {w n } is dense in V (here we use that V is separable in order to ensure existence of such a sequence). Let V m := span {w n : 1 n m}. As a finite dimensional vector space, the space V m is a closed subspace of V, H and V. It will be equipped with the norms coming from these three spaces. Note that the three norms are equivalent on V m.

15 3. THE THEOREM OF J.-L. LIONS 15 The restriction of the form a to the space V m (i.e. the form a m : V m V m R defined by a m (u, v) := a(u, v)) is a bilinear, bounded and elliptic form. Hence, there exists an operator A m : (V m, V ) (V m, V ) such that A m u, v V,V = a m (u, v) = a(u, v) for every u, v V m. Consider the ordinary differential equation (3.5) u m (t) + A m u m (t) = f m (t), t [, T], u m () = u m, where u m := P mu, P m : H H being the orthogonal projection in H onto V m, and where f m (t) = P m f (t) (note that the orthogonal projection P m extends to a bounded projection V V and that P m V = 1). The problem (3.5) is a linear inhomogeneous ordinary differential equation in a finite dimensional Hilbert/Banach space and we know from the theory of ordinary differential equations that (3.5) admits a unique solution u m C 1 ([, T]; V m ). Multiplying the equation (3.5) with u m, we obtain and hence, by ellipticity of a, ( u m (t), u m (t)) H + (A m u m (t), u m (t)) H = ( f m (t), u m (t)) H 1 d 2 dt u m(t) 2 H + η u m (t) 2 V u m (t), u m (t) V,V + A m u m (t), u m (t) V,V + ω u m (t) 2 H = f m (t), u m (t) V,V + ω u m (t) 2 H C η f m (t) 2 V + η 2 u m(t) 2 V + ω u m (t) 2 H. As a first consequence, we obtain the inequality 1 d 2 dt u m(t) 2 H C η f m (t) 2 V + ω u m(t) 2 H. By Gronwall s lemma, this implies for every t [, T], t u m (t) 2 H e 2ωt u m 2 H + C η e 2ω(t s) f m (s) 2 V ds C ( u 2 H + T f (s) 2 V ds), where C e 2ωT (C η + 1). When we plug this inequality into the above inequality, then we obtain 1 d 2 dt u m(t) 2 H + η 2 u m(t) 2 V C η f m (t) 2 V + C( u 2 H + This implies, when integrating over (, T), η 2 T u m (t) 2 V dt u m(t) 2 H C ( u 2 H + T T f (s) 2 V ds). f (s) 2 V ds).

16 16 2. LINEAR GRADIENT SYSTEMS This and the equation (3.5) imply T T T u m (t) 2 V dt A m u m (t) 2 V dt + f m (t) 2 V dt M T 2(MC + 1) η u m (t) 2 V dt + T f (t) 2 V dt ( T f (t) 2 V dt + u 2 H). Summing up, we see that there exists a constant C such that for every m 1 T T ( T ) u m (t) 2 V dt + u m (t) 2 V dt C f (t) 2 V dt + u 2 H. The right-hand side is finite by assumption and does not depend on m 1. As a consequence, (u m ) is bounded in L 2 (, T; V) and W 1,2 (, T; V ). By reflexivity, we can thus extract a subsequence (which we denote again by (u m )) such that This means that for every ϕ L 2 (, T; V ) T lim m and for every ϕ L 2 (, T; V) T lim m u m u in L 2 (, T; V) and u m v in L 2 (, T; V ). ϕ(t), u m (t) V,V = u m (t), ϕ(t) V,V = T T ϕ(t), u(t) V,V v(t), ϕ(t) V,V. Let w V be any fixed vector and let ϕ D(, T) be a scalar test function. Then an integration by parts yields T u(t) ϕ(t) dt, w V,V = T = lim u(t), ϕ(t)w V,V m T = lim = = T m T T Since this equality is true for every w V, we find that T u ϕ = T u m (t), ϕ(t)w V,V u m (t), ϕ(t)w V,V v(t), ϕ(t)w V,V v(t)ϕ(t), w V,V. vϕ in V,

17 3. THE THEOREM OF J.-L. LIONS 17 for every test function ϕ D(, T). Hence, by definition of the Sobolev space, the function u belongs to W 1,2 (, T; V ) and u = v. Since A : V V is a bounded linear operator, we find that i.e. for every ϕ L 2 (, T; V) Note that also T lim m Au m Au in L 2 (, T; V ), Au m (t), ϕ(t) V,V = T A m u m Au in L 2 (, T; V ). Au(t), ϕ(t) V,V. In order to see this, let w V n for some n 1 and let ϕ L 2 (, T). Then, for every m n, T A m u m (t), ϕ(t)w V,V = = = T T T T a m (u m (t), ϕ(t)w) a(u m (t), ϕ(t)w) Au m (t), ϕ(t)w V,V Au(t), ϕ(t)w V,V (m ). Since n V n is dense in V, and since therefore the set {ϕ( )v : ϕ L 2 (, T), v n V n } is total in L 2 (, T; V), the last claim follows. Note also that f m f in L 2 (, T; V ). We thus obtain for every ϕ L 2 (, T; V) T u(t), ϕ(t) V,V = lim m T = lim = T m T u m (t), ϕ(t) V,V f m (t) A m u m (t), ϕ(t) V,V f (t) Au(t), ϕ(t) V,V. Since this equality holds for every ϕ L 2 (, T; V), we find that u(t) + Au(t) = f (t) for a.e. t [, T], i.e. u is a solution of our differential equation. It remains to show that u verifies also the initial condition. Let w V and let ϕ C 1 ([, T]) be such that ϕ() = 1 and ϕ(t) =. Then an integration by parts yields on the one hand T u, ϕw V,V = u(), w V,V T u, ϕw V,V.

18 18 2. LINEAR GRADIENT SYSTEMS On the other hand, since u m u in H, T u, ϕw V,V = lim m T u m, ϕw V,V ( = lim um (), w V,V m = lim u m, w V,V m = u, w V,V Comparing both equalities, we obtain for every w V. Hence, u() = u. T T u(), w V,V = u, w V,V T u m, ϕw V,V u, ϕw V,V u, ϕw V,V. REMARK 3.6. Lions Theorem says that the operator A : V V, considered as a closed, unbounded operator on V with domain D(A) = V, has L 2 -maximal regularity. This follows when regarding the inhomogeneous problem with initial value u() =. Moreover, it follows from Lions Theorem, especially the solvability of the initial value problem, that H Tr 2 (V, V). Together with Lemma 3.4 this implies the identity Tr 2 (V, V) = H, i.e. a complete description of the trace space in this special situation. EXAMPLE 3.7. We consider the linear heat equation with Dirichlet boundary conditions and initial condition u t (t, x) u(t, x) = f (t, x) (t, x) T, (3.6) u(t, x) = x, u(, x) = u (x) x, where R n is any open set and T = (, T). This heat equation can be abstractly rewritten as a linear Cauchy problem u(t) + Au(t), t [, T], u() = u, where A : H 1 () H 1 () is the Dirichlet-Laplace operator associated with the form a : H 1 () H1 () R defined by a(u, v) = u v. It follows from Lions Theorem that for every f L 2 (, T; H 1 ()) and every u L 2 () there exists a unique solution u W 1,2 (, T; H 1 ()) L 2 (, T; H 1 ())

19 4. * L p -MAXIMAL REGULARITY 19 of this problem. A particular situation arises when = R n (in this case the boundary conditions are obsolete) and when f =, because in this case we have an explicit formula for the solution. Using the heat kernel, one has for every u L 2 (R n ) the solution u of the heat equation is given by 1 u(t, x) = e x y 2 /(4t) u (4πt) n/2 (y) dy. R n Lions Theorem implies that this solution belongs to the space u W 1,2 (, T; H 1 (R n )) L 2 (, T; H 1 (R n )) C([, T]; L 2 (R n )). 4. * L p -maximal regularity Let A : X D(A) X be a closed, linear, densely defined operator on X. We consider the abstract linear inhomogeneous Cauchy problem (4.1) u(t) + Au(t) = f (t), t [, T], u() =. Here, f L p (, T; X) for some 1 p. DEFINITION 4.1. (a) A function u C 1 ([, T]; X) C([, T]; D(A)) is called a classical solution if u() = and if u satisfies the differential equation (4.1) for every t [, T]. (b) A function u W 1,p (, T; X) L p (, T; D(A)) is called a (L p ) strong solution if u() = and if u satisfies the differential equation (4.1) for almost every t [, T]. DEFINITION 4.2. We say that A has L p -maximal regularity (on (, T)) if for every f L p (, T; X) there exists a unique strong solution u W 1,p (, T; X) L p (, T; D(A)) of the problem (4.1). By definition, if A has L p -maximal regularity, then the Cauchy problem (4.1) is uniquely solvable in the space W 1,p (, T; X) L p (, T; D(A)), for every f L p (, T; X). It will be convenient to introduce the maximal regularity space MR p (a, b; X, D(A)) := W 1,p (a, b; X) L p (a, b; D(A)) ( a < b ) which is naturally endowed with the norm u MRp := u W 1,p (a,b;x) + u L p (a,b;d(a)). Since W 1,p (a, b; X) and L p (a, b; D(A)) are Banach spaces, MR p (a, b; X, D(A)) is also a Banach space. If there is no danger of confusion, we will write MR p (a, b) instead of MR p (a, b; X, D(A)). We will first show that the definition of L p -maximal regularity is independent of T >, so that it suffices in fact to speak only of L p -maximal regularity. On the way we will also show that the initial value problem is uniquely solvable in the maximal regularity space, at least for certain initial values. For this, we first need the following locality lemma.

20 2 2. LINEAR GRADIENT SYSTEMS LEMMA 4.3. Assume that A has L p -maximal regularity on (, T). If f L p (, T; X) is zero on the interval (, T ) (with < T T), and if u MR p (, T; X, D(A)) is the corresponding solution of (4.1), then u = on (, T ). PROOF. Define the function f (t + T ) if t T T, g(t) := if T T < t T. Then g L p (, T; X). By definition of L p -maximal regularity, there exists a unique v MR p (, T; X, D(A)) solution of (4.1). Now define if t T, w(t) := v(t T ) if T < t T. Then the function w restricted to the two intervals [, T ] and [T, T] belongs to the maximal regularity spaces MR p (, T ) and MR p (T, T), respectively. Since w is also continuous in T (note that v() =!), we actually have w MR p (, T). It follows easily from the definition of w (the definition of g and v), that w solves the problem (4.1) for the function f. Since (4.1) is uniquely solvable, u = w, and therefore u = on [, T ]. We also have to define the trace space Tr p (X, D(A)) := {u() : u MR p (, 1)}, which is naturally a Banach space for the norm u Trp := inf{ u MRp (,1) : u MR p (, 1) and u() = u }. If there is no danger of confusion, we simply write Tr p instead of Tr p (X, D(A)). The space Tr p is called trace space since it contains all traces in t = of functions u MR p (, 1). Note that we can evaluate u() for every function u in the maximal regularity space MR p (, T; X, D(A)) since W 1,p (, T; X) is contained in the space of all continuous functions (see vector-valued Sobolev spaces in one dimension). Clearly, by definition, Tr p is contained in X, and since for every u D(A) the constant function u u belongs to MR p (, 1), one has the inclusions D(A) Tr p X. It turns out that Tr p is a strictly contained between D(A) and X (see below). For the moment, however, we need not to know this. LEMMA 4.4. The following are true: (a) For every T > and every t T one has (b) One has the inclusion Tr p = {u(t) : u MR p (, T)}. MR p (, T) C([, T]; Tr p )

21 4. * L p -MAXIMAL REGULARITY 21 and there exists a constant C (depending on T > ) such that u C([,T];Trp ) C u MRp for every u MR p. PROOF. The spaces MR p (, T) and MR p (, 1) are isomorphic via the isomorphism u u( T). Hence, for every T >, Tr p = {u() : u MR p (, T)}, and u Trp,T := inf{ u MRp (,T) : u MR p (, T) and u() = u } defines an equivalent norm on Tr p. Given u MR p (, T) we may define the extension v MR p (, 2T) by u(t) if t T, v(t) := u(2t t) if T < t 2T. We define next the functions u t MR p (, T) by u t (s) := v(t + s), s, t T. Then one sees that u(t) = u t () Tr p for every t T and since t u t, [, T] MR p [, T] is continuous, one obtains from the definition of the norm on Tr p that is continuous. Moreover, t u(t), [, T] Tr p sup u(t) Trp C sup u(t) Trp,2T C v MRp (,2T) = 2C u MRp (,T). t [,T] t [,T] THEOREM 4.5 (Initial value problem). Assume that A has L p -maximal regularity on (, T). Then for every u Tr p there exists a unique u MR p (, T) solution of the problem u(t) + Au(t) =, t [, T], u() = u. PROOF. Existence: Let u Tr p. By definition of Tr p and Lemma 4.4, there exists v MR p (, T) such that v() = u. By definition of L p -maximal regularity, there exists w MR p (, T) solution of ẇ(t) + Aw(t) = v(t) + Av(t), t [, T], w() =. Now put u := v w. Uniqueness: Let u and v be two solutions of the initial value problem. Then u v is a solution of the same initial value problem with initial value u replaced by. The solution for that problem, however, is unique by definition of L p -maximal regularity. Hence, u = v. THEOREM 4.6 (Independence of T > ). Assume that A has L p -maximal regularity on (, T). Then A has L p -maximal regularity on (, T ) for every T >.

22 22 2. LINEAR GRADIENT SYSTEMS PROOF. Fix T (, T]. Let f L p (, T ; X) and extend f by zero on (T, T]. The resulting function is denoted by f. Let ũ MR p (, T) be the unique solution of (4.1). Let u be restriction of ũ to the interval [, T ]. Then u MR p (, T ) is a solution of (4.1) with T replaced by T. Hence, we have proved existence of strong solutions. In order to prove uniqueness, by linearity, it suffices to show that u = is the only solution of (4.1) with T replaced by T and with f =. So let u be some solution of the problem u(t) + Au(t) =, t [, T ], u() =. REMARK 4.7. Theorem 4.6 allows us just to speak of L p -maximal regularity of an operator A or of the Cauchy problem (4.1) without making the T > precise. 5. * Interpolation and L p -maximal regularity The aim of this section is to study interpolation results for maximal regularity. In particular, as a corollary, we will prove that the operator A H : D(A H ) H associated with a bounded, elliptic bilinear form a : V V R has L 2 -maximal regularity. Given two Banach spaces X, Y such that Y X, and given T >, p [1, ], we define the maximal regularity space and the trace space MR p (, T; X, Y) := W 1,p (, T; X) L p (, T; Y) Tr p (X, Y) := {u() : u MR p (, T; X, Y)} with usual norms. The maximal regularity space and the trace space used up to now was obtained for Y = D(A). The definition of Tr p (X, Y) is independent of T >. LEMMA 5.1 (Interpolation of a bounded linear operator). Let X 1, X 2, Y 1, Y 2 be four Banach spaces such that Y i X i for i = 1, 2. Let S : X 1 X 2 be a bounded linear operator such that its restriction to Y 1 is a bounded linear operator S : Y 1 Y 2. Then, for every p [1, ], the restriction of S to Tr p (X 1, Y 1 ) is a bounded linear operator S : Tr p (X 1, Y 1 ) Tr p (X 2, Y 2 ) and S L(Trp (X 1,Y 1 ),Tr p (X 2,Y 2 )) max{ S L(X1,X 2 ), S L(Y1,Y 2 )}. Moreover, if S : X 1 X 2 and S : Y 1 Y 2 are invertible, then S : Tr p (X 1, Y 1 ) Tr p (X 2, Y 2 ) is invertible, too. PROOF. Let u Tr p (X 1, Y 1 ). By definition of the trace space, and by definition of the norm on the trace space, for every ε > there exists u MR p (, T; X 1, Y 1 ) such that u MRp (1 + ε) u Trp. Put v(t) := S u(t). Then v MR p (, T; X 2, Y 2 ) and v MRp = v W 1,p (,T;X 2 ) + v L p (,T;Y 2 ) S L(X1,X 2 ) u W 1,p (,T;X 1 ) + S L(Y1,Y 2 ) u L p (,T;Y 1 ) max{ S L(X1,X 2 ), S L(Y1,Y 2 )} u MRp <.

23 5. * INTERPOLATION AND L p -MAXIMAL REGULARITY 23 In particular, v() = S u() = S u Tr p (X 2, Y 2 ) and S u Trp (X 2,Y 2 ) v MRp max{ S L(X1,X 2 ), S L(Y1,Y 2 )} u MRp (1 + ε) max{ S L(X1,X 2 ), S L(Y1,Y 2 )} u Trp (X 1,Y 1 ). Since ε > was arbitrary, the first claim follows. If S : X 1 X 2 and S : Y 1 Y 2 are invertible, then one applies the above argument to the operator S 1 : X 2 X 1 whose restriction to Y 2 is a bounded linear operator S 1 : Y 2 Y 1. REMARK 5.2. The situation in the interpolation lemma. The boundedness of S : X 1 X 2 and S : Y 1 Y 2 is assumed, the boundedness of S in the interpolation spaces is a consequence: X 1 Tr p (X 1, Y 1 ) Y 1 The following lemma will not be proved. S X 2 S Tr p(x 2, Y 2 ) S Y 2 LEMMA 5.3. Let X, Y be two Banach spaces such that Y X. Then, for every p [1, ], Tr p (L p (, T; X), L p (, T; Y)) = L p (, T; Tr p (X, Y)). Let A : D(A) X be a closed linear operator on X. This implies that the domain D(A) equipped with the graph norm is a Banach space. We can define the restriction of A to the space D(A) by D(A 1 ) := {x D(A) : Ax D(A)}, A 1 x := Ax. This restriction is again a closed linear operator (exercice!). LEMMA 5.4. Let A : D(A) X be a closed linear operator on X and define A 1 : D(A 1 ) D(A) as above. Assume that A + ωi is invertible and that A has L p -maximal regularity. Then A 1 has L p -maximal regularity. PROOF. The operator A + ωi is an isomorphism between the Banach spaces D(A) and X, and also between the Banach spaces D(A 1 ) and D(A). Let f L p (, T; D(A)). Then (A + ωi) f L p (, T; X) and by L p -maximal regularity there exists a unique solution u MR p (, T; X, D(A)) of the problem u + Au = (A + ωi) f, t [, T], u() =. Multiply this differential equation by (A + ωi) 1 and put v(t) := (A + ωi) 1 u(t). Then v MR p (, T; D(A), D(A 1 )) is solution of the problem v + Av = f, t [, T], v() =.

24 24 2. LINEAR GRADIENT SYSTEMS This solution is unique since every solution in MR p (, T; D(A), D(A 1 )) is also a solution in MR p (, T; X, D(A)) of the same problem and the solution in the latter space is unique by L p -maximal regularity. As a consequence, A 1 : D(A 1 ) D(A) has L p -maximal regularity. REMARK 5.5. One can repeat the above argument and restrict the operator A to the space D(A 1 ) which is also a Banach space. This restriction is given by D(A 2 ) := {x D(A 1 ) : Ax D(A 1 )}, A 2 x := Ax, and it is also a closed linear operator. By iteration, one can define closed linear operators D(A k ) := {x D(A k 1 ) : Ax D(A k 1 )}, A k x := Ax, and one obtains the following picture: D(A) D(A 1 ) D(A 2 ). A X =: X A D(A) =: X 1 A D(A 1) =: X 2 If A + ωi is invertible and if A has L p -maximal regularity, then each operator A k has L p -maximal regularity. Even more is true: we know from the interpolation lemma (Lemma 5.1) that A is also a closed linear operator on the interpolation spaces between X and D(A). In the following theorem we prove that if A + ωi is invertible and if A has L p -maximal regularity, then also the restriction of A to Tr p (X, D(A)) has L p -maximal regularity. THEOREM 5.6. Let A : D(A) X be a closed linear operator on X and define its restriction to Tr p (X, D(A)) by D(A Trp ) := Tr p (D(A), D(A 1 )), A Trp x := Ax. Assume that A + ωi is invertible and that A has L p -maximal regularity. Then A Trp has L p -maximal regularity. PROOF. Define MR p(, T; X, D(A)) := {u MR p (, T; X, D(A)) : u() = }.

25 and define the operator 5. * INTERPOLATION AND L p -MAXIMAL REGULARITY 25 S : MR p(, T; X, D(A)) L p (, T; X), u u + Au. The operator S is clearly bounded. Moreover, the operator A has L p -maximal regularity if and only if the operator S is invertible. Hence, by assumption, S is invertible. The restriction of S to the space MR p (, T; D(A), D(A 1 )) is a bounded operator with values in L p (, T; D(A)), and, by Lemma 5.4, this restriction is also invertible. By Lemma 5.3, we have and Tr p (L p (, T; D(A)), L p (, T; D(A 1 ))) = L p (, T; Tr p (D(A), D(A 1 ))), It then follows that Tr p (W 1,p (, T; X); W 1,p (, T; D(A))) = W 1,p (, T; Tr p (X, D(A))). Tr p (MR p (, T; X, D(A)), MR p (, T; D(A), D(A 1 ))) = = MR p (, T; Tr p (X, D(A)), Tr p (D(A), D(A 1 ))). By the Interpolation Lemma (Lemma 5.1), the restriction S : MR p(, T; Tr p (X, D(A)), Tr p (D(A), D(A 1 ))) L p (, T; Tr p (X, D(A))), u u + Au. is bounded and invertible. This means that the operator A Trp has L p -maximal regularity. REMARK 5.7. One has the equality Tr p (D(A), D(A 1 )) = {x D(A) : Ax Tr p (X, D(A))}, so that A Trp is really the restriction of A to the space Tr p (X, D(A)). In order to prove this equality, let u Tr p (D(A), D(A 1 )) D(A). Then there exists u MR p (, T; D(A), D(A 1 )) such that u() = u. Put v(t) := (A+ωI)u(t). Then v MR p (, T; X, D(A)) and thus (A + ωi)u Tr p (X, D(A)). Hence, u D(A Trp ). The other inclusion is proved similarly, using the invertibility of A + ωi. REMARK 5.8. As before, the procedure of considering restrictions to intermediate spaces can be repeated on the smaller spaces X 1 = D(A), X 2 = D(A 1 ), etc.. One thus

26 26 2. LINEAR GRADIENT SYSTEMS obtains the following picture: D(A) Tr p (D(A), D(A 1 )) D(A 1 ) Tr p (D(A 1 ), D(A 2 )) D(A 2 ). A X =: X A Tr p (X, D(A)) =: X 1/p A D(A) =: X 1 A Tr p(d(a), D(A 1 )) =: X 1+1/p A D(A 1 ) =: X 2 If A + ωi is invertible and if A has L p -maximal regularity, then each operator in this picture has L p -maximal regularity. COROLLARY 5.9. Let a : V V R be a bilinear, bounded, elliptic form and let A H : D(A H ) H be the associated operator on H. Then A H has L 2 -maximal regularity. In particular, for every f L 2 (, T; H) and every u V there exists a unique solution u W 1,2 (, T; H) L 2 (, T; D(A H )) of the problem u(t) + Au(t) = f (t), t [, T], u() = u. PROOF. By Lions Theorem (Theorem 3.1, see also Remark 3.6), the operator A : V V associated with the form a has L 2 -maximal regularity. By ellipticity of the form a and by the theorem of Lax-Milgram, the operator A + ωi is invertible. Hence, by Theorem 5.6, the restriction of A to the trace space Tr 2 (V, V) has L 2 -maximal regularity, too. But by Remark 3.6, this trace space is equal to H and the restriction of A to the space H is nothing else than A H. Hence, A H has L 2 -maximal regularity. For the second statement, one has to prove that Tr 2 (H, D(A H )) = V. EXAMPLE 5.1. We consider again the linear heat equation (3.6) with Dirichlet boundary conditions and initial condition from Example 3.7. From the results in this section follows that for every u H 1 () and every f L2 (, T; L 2 ()) the heat equation (3.6) admits a unique solution u W 1,2 (, T; L 2 ()) L 2 (, T; D( L2 )) C([, T]; H 1 ()), where D( L 2) is the domain in L 2 () of the Laplace operator with Dirichlet boundary conditions. If = (a, b) is a bounded interval, then D( L 2) = H 2 (a, b) H 1 (a, b) (exercice). One also has D( L 2) = H 2 () H 1 () if RN has smooth boundary, but this result is more difficult to prove and will be omitted..

27 5. * INTERPOLATION AND L p -MAXIMAL REGULARITY 27 Note that one can identify the spaces L 2 (, T; L 2 ()) and L 2 ((, T) ) in a natural way so that the inhomogeneity f is actually a real valued function on the product (, T) as suggested in the heat equation (3.6).

28

29 CHAPTER 3 Nonlinear gradient systems 1. Quasilinear equations: existence and uniqueness of local solutions Let X and D be two Banach spaces such that D is densely and continuously embedded into X. Fix 1 < p <. Let u Tr p (X, D) and f L p (, 1; X). Let, moreover, A : D L(D, X) and F : D X be two functions having the property that (H) for every T > and every u, v MR p (, T; X, D) one has A(u)v L p (, T; X) and F(u) L p (, T; X). In this section, we consider the quasilinear problem u + A(u)u + F(u) = f, t, (1.1) u() = u. A local solution of this problem will be a function u MR p (, T; X, D) which satisfies the differential equation almost everywhere on [, T] and which satisfies the initial condition. THEOREM 1.1 (Existence and uniqueness). Assume that there exists z MR p (, 1; X, D) such that z() = u and (i) there exists r >, L such that for every < T 1 and every u, v, w MR p (, T; X, D) satisfying u() = v() = w() = u and u z MRp, v z MRp, w z MRp r one has (A(u) A(v))w L p (,T;X) L u v MRp (,T) w MRp (,T), (ii) there exists r > and L T such that lim T L T = and for every < T 1 and every u, v MR p (, T; X, D) satisfying u() = v() = u and u z MRp, v z MRp r one has F(u) F(v) L p (,T;X) L T u v MRp (,T), (iii) for every < T 1, every g L p (, T; X) and every v Tr p the linear problem v + A(z)v = g, t [, T], v() = v admits a unique solution v MR p (, T; X, D). Then the quasilinear problem (1.1) admits a unique local solution u MR p (, T; X, D). 29

30 3 3. NONLINEAR GRADIENT SYSTEMS PROOF. For every T > we set M T := {u MR p (, T; X, D) : u() = u }. The set M T will be equiped with the metric induced by the norm in MR p. Functions in M already satisfy the initial condition from (1.1). Consider the nonlinear map R : M L p (, T; X), u (A(z) A(u))u F(u) + f. By the hypothesis (H), the map R is well defined. Consider also the solution map S : L p (, T; X) M, g S g := u, which assigns to every function g L p (, T; X) the unique solution in problem v + A(z)v = g(t), t [, T], v() = u. M of the By assumption (iii), this solution map is well defined, too. By definition of the two maps above, a function u MR p (, T; X, D) is a solution of the quasilinear problem (1.1) if and only if u M and S Ru = u, i.e. if u is a fixed point of S R. We have thus reduced the problem of existence to a fixed point problem which we will solve by using Banach s fixed point theorem. Let S : L p (, T; X) MR p (, T; X, D) be the solution operator which assigns to every function g L p (, T; X) the unique solution v := S g of the problem v + A(z)v = g, t [, T], v() =. There exists a constant C S independent of < T 1 such that S C S for every < T 1. We may assume that the constant r > from assumptions (i) and (ii) is the same. Let r > be such that r 1 min{r, 1 C S L }, choose < T 1 sufficiently small so that (1.2) L T 1 5 C S, z MRp (,T) r, and F(z) L p (,T;X) + ż + A(z)z L p (,T;X) + f L p (,T;X) 3 5 C S r. Such a parameter T clearly exists, by the assumption that lim T L T = and by the properties of the norms in L p and W 1,p. Set M := {u M T : u z MRp (,T) r }.

31 1. EXISTENCE AND UNIQUENESS OF LOCAL SOLUTIONS 31 The set M is a complete metric space for the metric induced by the norm MRp = MRp (,T). For every u M one has u MRp u z MRp + z MRp 2r. We prove that S R maps the set M into itself. In order to see this, let u M. Then S Ru z MRp = S Ru S (ż + A(z)z) MRp = S (Ru ż + A(z)z) MRp ( C S (A(z) A(u))u L p (,T;X) + F(z) F(u) L p (,T;X) + ) + F(z) L p (,T;X) + ż + A(z)z L p (,T;X) + f L p (,T;X) C S ( L z u MRp u MRp + L T z u MRp C S r ) C S ( 2Lr r + L T r C S r ) r ( ) = r. This proves that S Ru M. We next prove that S R is a strict contraction. In order to see this, let u, v M. Then S Ru S Rv MRp = S (Ru Rv) MRp ( C S (A(z) A(u))(u v) L p (,T;X) + (A(u) A(v))v L p (,T;X) + ) + F(u) F(v) L p (,T;X) ( ) C S L u z MRp + L v MRp + L T u v MRp C S (L r + L 2r + L T ) u v MRp ( ) u v MR p = 1 2 u v MR p. Hence, S R : M M is a strict contraction. By Banach s fixed point theorem, there exists a unique fixed point u M which by construction of S and R is a solution of the quasilinear problem (1.1). REMARK 1.2. It follows from the proof of Theorem 1.1 that one could actually also study non-autonomous (i.e. time-dependent) quasi-linear problems of the form (1.3) u + A(t, u)u + F(t, u) =, t, u() = u. Here A : [, T] D L(D, X) and F : [, T] D X are two functions such that (1.4) for every T > and every u, v MR p (, T) one has A(t, u)v L p (, T; X) and F(t, u) L p (, T; X). By this hypothesis, for every T >, the operators A : MR p (, T) MR p (, T) L p (, T; X) and F : MR p (, T) L p (, T; X) given respectively by (u, v) A(t, u)v

32 32 3. NONLINEAR GRADIENT SYSTEMS and (u, v) F(t, u) are well-defined. Theorem 1.1, with an obvious small change in condition (iii), holds then true for the non-autonomous problem 1.3, too. The following lemma gives sufficient conditions when the conditions (i) and (ii) of Theorem 1.1 are satisfied. LEMMA 1.3. Fix 1 < p <. The following are true: (i) If A : Tr p L(D, X) is Lipschitz continuous in a neighbourhood of u (with respect to the topology in Tr p ), then the assumption (i) of Theorem 1.1 is satisfied. (ii) If F : Tr p X is Lipschitz continuous in a neighbourhood of u (with respect to the topology in Tr p ), then the assumption (i) of Theorem 1.1 is satisfied. In order to prove this lemma, we need the following lemma. LEMMA 1.4. For every T > and every u MR p (, T; X, D) satisfying u() = one has (1.5) u C([,T];Trp ) 2 u MR p (,T;X,D). PROOF. Every function u MR p (, T; X, D) satisfying u() = can be extended to a function ū MR p (, ; X, D) by setting u(t) if t T, ū(t) := u(2t t) if T t 2T, if 2T t, and for this particular extension one has ū MRp (, ;X,D) 2 u MRp (,T;X,D). Note that in this reasoning it is important that u() =! As a consequence, by definition of the norm in the trace space, for every t, and the inequality (1.5) follows. u(t) Trp ū(t + ) MRp (,1;X,D) 2 u MRp (,T;X,D), PROOF OF LEMMA 1.3. (i) By assumption, there exists r > and L such that A(u) A(v) L(D,X) L u v Trp, whenever u, v Tr p are such that u u Trp r and v u Trp r. Let z MR p (, 1) be such that sup t [,1] z(t) u Trp r/3. For every u, v, w MR p (, T) with u() = v() = w() = u and u z MRp r/3, v z MRp r/3 and w z MRp r/3 we have, by Lemma 1.4, and therefore, for every t [, 1], u z C([,1];Trp ), v z C([,1];Trp ) 2r/3, u(t) u Trp, v(t) u Trp r.

33 As a consequence, 1. EXISTENCE AND UNIQUENESS OF LOCAL SOLUTIONS 33 (A(u) A(v))w L p (,T;X) sup A(u(t)) A(v(t)) L(D,X) w L p (,T;D) t [,T] L u v C([,T];Trp ) w MRp 2L u v MRp w MRp. Hence, A satisfies the condition (i) of Theorem 1.1. (ii) By assumption, there exists r > and L such that F(u) F(v) X L u v Trp, whenever u, v Tr p are such that u u Trp r and v u Trp r. Similarly as above, for every u, v MR p (, T) with u() = v() = u and u z MRp r/3 and v z MRp r/3 we obtain F(u) F(v) L p (,T;X) T 1 p F(u) F(v) C([,T];X) T 1 p L u v C([,T];Trp ) 2T 1 p L u v MRp. Hence, F satisfies the condition (ii) of Theorem 1.1. A special case of the quasilinear equation (1.1) is obtained when the function A is constant. In this case, we call the quasilinear equation semilinear. We will formulate the local existence and uniqueness of solutions in this special case. For this, we assume again that D and X are two Banach spaces such that D is densely and continuously embedded into X. Let A : D X be a fixed bounded linear operator. Fix 1 < p < and let F : D X be a function such that (1.6) for every T > and every u MR p (, T; X, D) one has F(u) L p (, T; X). We consider the semilinear problem u + Au + F(u) = f, t, (1.7) u() = u, where f L p (, T; X) and u Tr p = Tr p (D, X). A local solution of this problem will be a function u MR p (, T; X, D) which satisfies the differential equation almost everywhere on [, T] and which satisfies the initial condition. Note that the linear operator A gives rise to the constant function A : D L(D, X) which assigns to every u D the operator A. A constant function A clearly satisfies condition (i) from Theorem 1.1. Hence, the following result is an immediate corollary to Theorem 1.1. Note that condition (i) in the following corollary is nothing else than condition (ii) from Theorem 1.1. COROLLARY 1.5 (Existence and uniqueness). Assume that there exists z MR p (, 1; X, D) such that z() = u and

34 34 3. NONLINEAR GRADIENT SYSTEMS (i) there exists r > and L T such that lim T L T = and for every < T 1 and every u, v MR p (, T; X, D) satisfying u() = v() = u and u z MRp, v z MRp r one has F(u) F(v) L p (,T;X) L T u v MRp (,T), (ii) for every g L p (, 1; X) and every v Tr p the linear problem v + Av = g, t [, 1], v() = v admits a unique solution v MR p (, 1; X, D), that is, the operator A has L p -maximal regularity. Then the semilinear problem (1.7) admits a unique local solution u MR p (, T; X, D). EXAMPLE 1.6 (Semilinear heat equation). Let R N (N 3) be open, T >, T = (, T), f C 1 (R), and consider the semilinear heat equation with Dirichlet boundary conditions and initial condition: (1.8) u t (t, x) u(t, x) + f (u(t, x)) = (t, x) T, u(t, x) = x, u(, x) = u (x) x, Assume in addition that there exists some constant C such that 2 f (s) s N 2 for every s R. COROLLARY 1.7. For every u H 1 () there exists a unique local solution u W 1,2 (, T ; L 2 ()) L 2 (, T ; D( L 2)) C([, T ]; H 1 ()) of the problem (3.6). PROOF. In fact, we may apply Corollary 1.5, where A = L 2 is the Dirichlet- Laplace operator associated with the form a : H 1 () H1 () R given by a(u, v) = u v (which has L2 -maximal regularity on L 2 () by Corollary 5.9) and where F : H 1 () L2 () is the Nemytski operator associated with the function f : F(u)(x) := f (u(x)). By Lemma 1.3, it suffices to show that this Nemytski operator is locally Lipschitz continuous. We will need the Sobolev embedding H 1 () Lq (),

35 1. EXISTENCE AND UNIQUENESS OF LOCAL SOLUTIONS 35 which is true for q = 2N. From this embedding, the growth condition on f, the N 2 mean value theorem and Hölder s inequality we deduce that for every u, v H 1 () F(u) F(v) 2 = f (u) f (v) 2 L 2 = f (ξ(x))(u(x) v(x)) 2 4 ξ(x) N 2 u v 2 4 u v N 2 u v 2 ( ) u v N 2 2N 2 N ( ) u v 2N N 2 N 2 N max{ u H 1, v H 1 } 4 N 2 u v 2 H 1. Hence, for every R > there exists a Lipschitz constant L such that for every u, v H 1 () with norms less than R one has F(u) F(v) L 2 L u v H 1. In fact, one may take L := R N 2 2. In other words, F is Lipschitz continuous on bounded subsets of H 1 (). EXAMPLE 1.8 (Cahn-Hilliard equation). Let R 3 be a bounded domain which is regular in the send that the domain of the Dirichlet-Laplace operator A = A L 2 which is associated with the form a : H 1 () H1 () R, a(u, v) = u v, is given by D(A) = H 2 H 1 (). In this example, we consider the Cahn-Hilliard equation u t (t, x) + ( u(t, x) f (u(t, x))) = (t, x) T, (1.9) u(t, x) = u(t, x) = x, u(, x) = u (x) x, where as before T = (, T), and where the nonlinearity f belongs to C 3 (R). No growth restrictions on f are imposed. We will apply Corollary 1.5 in order to prove existence and uniqueness of solutions of the Cahn-Hilliard equation, at least for initial values u H 2 H 1 (). For this, we start by the following lemma. LEMMA 1.9. The bilinear form b : H 2 H 1 () H2 H 1 () R defined by b(u, v) = u v is bounded and elliptic.