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Partial Differential Equations II

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Partial Differential Equations II Partial differential equations II Radim Cajzl Dalibor Pražák MFF UK, 2016 Introduction These lecture notes are a transcript of the course “PDE2”, given by the second author at MFF UK during spring 2016. All the material presented here is well-known. Precise wording of theorems and their proofs were inspired by various sources we list in the bibliography. An up-to-date version of these notes, along with supplementary material, can be found at the webpage of the course: http://www.karlin.mff.cuni.cz/ prazak/vyuka/Pdr2/. Feel free to write an e-mail to the authors in∼ the case you would like to share some ideas or report found errors in the text. The authors’ email adresses are [email protected] and [email protected]. Special thanks go to Zdenek Mihula for providing detailed solutions to selected exercises. 2 Contents Introduction 2 Chapter 1. Vector–valued functions 5 1.1. Vector–valued integrable functions (Bochner Integral) 5 1.2. AC functions and weak time derivative 8 1.3. Geometry and duality in Lp (I; X) spaces 10 1.4. More on weakly differentiable functions: extensions, approximations, embeddings 13 Chapter 2. Parabolic second order equations 18 Definition of weak solution 19 Uniqueness of weak solution 20 Compactness of weak solutions 22 Existence of weak solutions 24 Maximum principle for weak solutions 27 Existence of strong solution to the heat equation 28 Chapter 3. Hyperbolic second order equations 30 Definition of weak solution 30 Uniqueness of weak solution 31 Time derivative as a test function 32 Existence of weak solution 33 Existence of strong solution 35 Finite speed of propagation 36 Chapter 4. Semigroup theory 38 4.1. Homogeneous equation 38 c0 semigroup 38 Generator of semigroup, properties 39 Uniqueness of the semigroup 41 Definintion: resolvent, resolvent set, spectrum 41 Resolvent and Laplace transform 41 Hille–Yosida theorem for contractions 42 4.2. Nonhomogeneous equation 44 Hille’s characterization of L1(I; D(A)) 44 Definition: mild solution 44 Properties of semigroup convolutions 45 Equivalent definition of mild solution 45 Regularity of mild solution 47 Remarks on exponential formulas 47 Applications of semigroup theory to heat and wave equation 48 3 CONTENTS 4 Excercises 51 Hints 55 Solutions 57 Bochner integral, convolution 57 Relation of Lp (I; Lp(Ω)) and Lp(I Ω) 60 Integral formulation of weak derivative× with initial condition 60 Gelfand triple and embedding L2 in W −1;2 62 Continuity of projection to laplacian eigenvalues 63 Characterization of semigroups with bounded generators 65 Semigroup generated by the heat equation 65 Semigroup of shift operators 66 Uniqueness of semigroup w.r.t. generator 72 Bibliography 74 Appendices 75 Properties of weak convergence 76 Time derivative of integral over time-dependent domain 77 Regularity of the heat equation 78 CHAPTER 1 Vector–valued functions Vector–valued . values in infinite–dimensional Banach space. Notation: we consider: u (t): I X, I = [0;T ] is time interval, X is Banach space, • ! u X is norm in X • k ∗k ∗ ∗ X is dual space X , x ; x ∗ is notation of duality • h iX ;X scalar case: X = R • 1.1. Vector–valued integrable functions (Bochner Integral) Def.: [Simple, weakly and strongly measurable function]. Function u (t): I X is called ! PN simple: if u (t) = χAj (t) xj, where Aj I are (Lebesgue in R) measurable and xj X j=1 ⊂ 2 (strongly) measurable: if there exist simple functions un (t) such that un (t) u (t), n for a. e. t I. ! ! 1 weakly2 measurable: if the (scalar) function t x∗; u (t) is measurable in I for x∗ X∗. 7! h i 8 2 Remark: (1) Strongly meas. = weakly meas. (2) u (t) simple )u (t) is measurable and u (I) X is finite. () ⊂ Theorem 1.1 [Pettis characterization of measurability]. Function u (t): I X is measurable iff u (t) is weakly measurable and moreover N I such that λ (N) = 0 and u (I N) is separable! (i. e., u (t) is “essentially separably–valued”). 9 ⊂ n Corollary. (1) For X separable: weakly measurable strongly measurable. () (2) un (t) is measurable, un (t) u (t) a. e. = u (t) is measurable. (3) u (t) continuous = u (t) measurable! ) ) th.1.1 Proof: un (t) measurable = Nn I, λ (Nn) = 0. Set u (I Nn) = Mn X is separable, also N0 I, ) 9 ⊂ n ⊂ 9 ⊂ 1 nX λ (N0) such that un (t) u (t) for t I N0. Set N = j=0Nj, then λ (N) = 0 and moreover u (I N) nM . separability is preserved! under8 countable2 n unions and[ closures = u (t) is essentially separably–valued.n ⊆ [ ∗ ∗ ) ∗ Is u (t) weakly measurable? Consider x X fixed, then the mapping t x ; un (t) is measurable and ∗ ∗ 2 ∗ 7! h i x ; un (t) x ; u (t) as n for a. e. t I, the mapping t x ; u (t) is measurable by scalar (Lebesgue) theory.h i ! h i ! 1 2 7! h i (3) HW1 Def.: [Bochner integral]. function u (t): I X is called Bochner integrable, if there exists un (t) simple R ! such that u (t) un (t) dt 0, n . The integral (Bochner integral) is defined as follows I k − k ! ! 1 R PN (1) u (t) dt = j=1 λ (Aj) xj, if u (t) is simple RI R (2) I u (t) dt = limn!1 I un (t) dt, if u (t) is integrable with un (t) simple from the definition above. Remark: It is possible to show that the definiton is correct, i. e., independent on the choice of Aj, xj, un (t). R R Also it is possible to prove that u (t) dt u (t) dt. I X ≤ k kX 5 1.1. VECTOR–VALUED INTEGRABLE FUNCTIONS (BOCHNER INTEGRAL) 6 ψ (t) 0 1 1 0 1 t − Figure 1.1.1. Regularization kernel Theorem 1.2 [Bochner characterization of measurability]. Function u (t): I X is Bochner inte- grable iff u (t) is measurable and R u (t) dt < . ! I k k 1 Theorem 1.3 [Lebesgue]. Let un (t): I X be measurable, un (t) u (t) as n for a. e. t I, let ! ! !R 1 R 2 there g (t): I R integrable and such that un (t) g (t) for n and a. e. t I. Then un (t) dt u (t) dt, 9 R ! k k ≤ 8 2 I ! I moreover un (t) u (t) dt 0 for n and u is integrable. I k − k ! ! 1 1 R h Recall: for scalar u (t): I R we say that t I is a Lebesgue point if limh!0 u (t + s) u (t) ds = 0. ! 2 2h −h j − j Lebesgue theorem: u (t): I R is integrable = a. e. t I is a Lebesgue point. ! ) 2 Def.: [Lebesgue point of function]. We say that t I is a Lebesgue point of u (t): I X if 2 ! 1 Z h u (t + s) u (t) ds 0 as h 0 . 2h −h k − k ! ! Theorem 1.4 [On Lebesgue points a. e.] If u (t): I X is integrable, then a. e. t I is a Lebesgue point. ! 2 Remark. Let u (t): I X be integrable. ! R t HW1 0 (1) Set U (x) = u (t) dt, t0 I fixed = U (t) = u (t) for every Lebesgue point of u (t), in particular a. t0 e. 2 ) (2) (regularization kernels) Let 0 (t): R R be bounded, zero outside interval [ 1; 1] and such that R 1 ! − −1 0 (s) ds = 1 (& possibly other regularizations). R 1 1 n (t) = n 0 (nt), u n (t) = u (t s) n (s) ds for t ;T or set u (t) = 0 outside I. ∗ R − 2 n − n Claim: u n (t) u (t) if t I is Lebesgue point of u. Proof: ∗ ! 2 Z u n (t) u (t) = u (t s) n (s) ds u (t) ∗ − R − − 1 Z n = [u (t s) u (t)] n (s) ds 1 − − − n 1 Z n (∗) = u n (t) u (t) c n u (t s) u (t) ds 0; n ) k − k ≤ |{z} − 1 k − k ! ! 1 −1 n h |{z} −h where ( ) holds by definition of Lebesgue points. ∗ X Proof of th. 1.4: By th. 1.1 there N0 I, λ (N0) = 0 such that u (I N0) x1; x2; : : : ; xk;::: . Set 9 ⊂ n ⊆ f g 'k = u (t) xk . scalar, integrable. By scalar version of the theorem Nk I, λ (Nk) = 0 such that 'k (t) k − k 1 9 ⊂ has Lebesgue point at every t I Nk. Set N = Nk, then λ (N) = 0 and t I N is Lebesgue point 2 n [k=0 8 2 n of u (t), i. e., 1 R h u (t + s) u (t) ds 0, h 0. Fix t I N, " > 0 arbitrary. Observe: k such that 2h −h k − k ! ! 2 n 9 1.1. VECTOR–VALUED INTEGRABLE FUNCTIONS (BOCHNER INTEGRAL) 7 u (t) xk ". Then k − k ≤ 1 Z h u (t + s) u (t) xk ds 2h −h k − ± k ≤ 1 Z h 1 Z h u (t + s) xk ds + u (t) xk ds 2h −h |k {z − k} 2h −h |k {z− k} '(t+s) ≤" | {z } | {z } I(h) ≤" (∗) 1 Z h I (h) = 'k (t + s) ds 'k (t); h 0 2h −h ! | {z } ! <" by choice of k since t is L. point of ' (t). ( ) < 2" for h small enough. ∗ Def.: [Spaces Lp (I; X)]. for p [1; ) we set 2 1 Z p p L (I; X) = u (t): I X measurable, u (t) X dt < ! I k k 1 L1 (I; X) = u (t): I X measurable, essentially bounded, ! i.
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