Tools and Problems in Partial Differential Equations

This textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations. Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory. Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject. Introduction Acknowledgements Contents Part I Tools and Problems 1 Elements of Functional Analysis and Distributions 1.1 Fréchet Spaces 1.2 Elements of Functional Analysis 1.2.1 Fixed Point Theorems 1.2.2 The Banach Isomorphism Theorem 1.2.3 The Closed Graph Theorem 1.2.4 The Banach–Steinhaus Theorem 1.2.5 The Banach–Alaoglu Theorem 1.2.6 The Ascoli Theorem 1.2.7 The Hahn–Banach Theorem 1.2.8 Hilbert Spaces 1.2.9 Spectral Theory of Self-Adjoint Compact Operators 1.2.10 Lp Spaces, 1 ≤p ≤+∞ 1.2.11 The Hölder and Young Inequalities 1.2.12 Approximation of the Identity 1.3 Elements of Distribution Theory 1.3.1 Distributions 1.3.2 Tempered Distributions 1.3.3 The Fourier Transform 1.3.4 The Stationary Phase Method 2 Statements of the Problems of Chap.1 3 Functional Spaces 3.1 Sobolev Spaces 3.1.1 Sobolev Spaces on Rd, d ≥1 3.1.1.1 Definition and First Properties 3.1.1.2 Density 3.1.1.3 Operations on Hs(Rd) 3.1.1.4 Sobolev Embeddings 3.1.1.5 Duality 3.1.1.6 Compactness 3.1.1.7 Traces 3.1.1.8 Equivalent Norm 3.1.2 Local Sobolev Spaces Hsloc(Rd) 3.1.3 Sobolev Spaces on an Open Subset of Rd 3.1.3.1 Definition and First Properties 3.1.3.2 Extension 3.1.3.3 Density 3.1.3.4 Poincaré Inequality 3.1.3.5 Sobolev Embedding 3.1.3.6 Duality 3.1.3.7 Compactness 3.1.3.8 Traces 3.1.4 Sobolev Spaces on the Torus 3.2 The Hölder Spaces 3.2.1 Hölder Spaces of Integer Order 3.2.1.1 Definition 3.2.1.2 Properties 3.2.2 Hölder Spaces of Fractional Order 3.2.2.1 Definition 3.2.2.2 Properties 3.3 Characterization of Sobolev and Hölder Spaces in Dyadic Rings 3.3.1 Characterization of Sobolev Spaces 3.3.2 Characterization of Hölder Spaces 3.3.3 The Zygmund Spaces 3.4 Paraproducts 3.5 Some Words on Interpolation 3.6 The Hardy–Littlewood–Sobolev Inequality 4 Statements of the Problems of Chap.3 5 Microlocal Analysis 5.1 Symbol Classes 5.1.1 Definition and First Properties 5.1.2 Examples 5.1.3 Classical Symbols 5.2 Pseudo-Differential Operators 5.2.1 Definition and First Properties 5.2.2 Kernel of a DO 5.2.3 Image of a DO by a Diffeomorphism 5.2.4 Symbolic Calculus 5.2.4.1 Composition 5.2.4.2 Adjoint 5.2.5 Action of the DO on Sobolev Spaces 5.2.6 Garding Inequalities 5.2.6.1 The Weak Inequality 5.2.6.2 The ``Sharp Garding'' Inequality 5.2.6.3 The Fefferman–Phong Inequality 5.3 Invertibility of Elliptic Symbols 5.4 Wave Front Set of a Distribution 5.4.1 Definition and First Properties 5.4.2 Wave Front Set and DO 5.4.3 The Propagation of Singularities Theorem 5.4.3.1 Bicharacteristics 5.4.3.2 The Propagation Theorem 5.5 Paradifferential Calculus 5.5.1 Symbols Classes 5.5.2 Paradifferential Operators 5.5.3 The Symbolic Calculus 5.5.4 Link with the Paraproducts 5.6 Microlocal Defect Measures 6 Statements of the Problems of Chap.5 7 The Classical Equations 7.1 Equations with Analytic Coefficients 7.1.1 The Cauchy–Kovalevski Theorem 7.1.1.1 The Linear Version 7.1.1.2 The Nonlinear Version 7.1.2 The Holmgren Uniqueness Theorem 7.2 The Laplace Equation 7.2.1 The Mean Value Property 7.2.2 Hypoellipticity: Analytic Hypoellipticity 7.2.3 The Maximum Principles 7.2.4 The Harnack Inequality 7.2.5 The Dirichlet Problem 7.2.5.1 Case g=0 7.2.5.2 Case g 0. 7.2.6 Spectral Theory 7.2.6.1 Weyl Law 7.2.6.2 Estimates of the Eigenfunctions 7.3 The Heat Equation 7.3.1 The Maximum Principle 7.3.2 The Cauchy Problem 7.4 The Wave Equation 7.4.1 Homogeneous Cauchy Problem 7.4.1.1 Properties of the Solution 7.4.1.2 Decay at Infinity 7.4.1.3 Finite Speed of Propagation 7.4.1.4 Huygens Principle 7.4.1.5 Influence Domain 7.4.1.6 Conservation of the Energy 7.4.1.7 Strichartz Estimates 7.4.2 Inhomogeneous Cauchy Problem 7.4.2.1 Finite Speed of Propagation 7.4.2.2 Strichartz Estimates 7.4.3 The Mixed Problem 7.5 The Schrödinger Equation 7.5.1 The Cauchy Problem 7.5.1.1 The Homogeneous Cauchy Problem 7.5.2 Properties of the Solution 7.5.2.1 Expression of the Solution 7.5.2.2 Infinite Speed of Propagation 7.5.2.3 Decay at Infinity of the Solution 7.5.2.4 Strichartz Inequality 7.5.2.5 The Inhomogeneous Problem 7.5.2.6 Nonhomogeneous Strichartz Inequality 7.6 The Burgers Equation 7.7 The Euler Equations 7.7.1 The Incompressible Euler Equations 7.7.1.1 Incompressibility 7.7.1.2 The Vorticity 7.7.1.3 Classical Solutions: The Lichtenstein Theorem 7.7.1.4 Weak Solutions: The Yudovitch Theorem 7.7.2 The Compressible Euler Equations 7.8 The Navier–Stokes Equations 7.8.1 Weak Solutions 7.8.2 The Leray Theorem (1934) 7.8.3 Strong Solutions: Theorems of Fujita–Kato and Kato 8 Statements of the Problems of Chap.7 Part II Solutions of the Problems and Classical Results 9 Solutions of the Problems 10 Classical Results 10.1 Some Classical Formulas 10.1.1 The Leibniz Formula 10.1.2 The Taylor Formula with Integral Reminder 10.1.3 The Faa–di-Bruno Formula 10.2 Elements of Integration 10.2.1 Convergence Theorems 10.2.2 Change of Variables in Rd 10.2.3 Polar Coordinates in Rd 10.2.4 The Gauss–Green Formula 10.2.5 Integration on a Graph 10.3 Elements of Differential Calculus 10.4 Elements of Differential Equations 10.4.1 The Precise Cauchy–Lipschitz Theorem 10.4.2 The Cauchy–Arzela–Péano Theorem 10.4.3 Global Theory 10.4.4 The Gronwall Inequality 10.5 Elements of Holomorphic Functions References Index