LEADER 05157nam 2200577 450 001 9910624310303321 005 20230508101631.0 010 $a9783031142680$b(electronic bk.) 010 $z9783031142673 035 $a(MiAaPQ)EBC7133438 035 $a(Au-PeEL)EBL7133438 035 $a(CKB)25299349700041 035 $a(PPN)266352650 035 $a(EXLCZ)9925299349700041 100 $a20230319d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 00$aResearch in PDEs and related fields $ethe 2019 Spring School, Sidi Bel Abbe?s, Algeria /$fKai?s Ammari, editor 210 1$aCham, Switzerland :$cBirkha?user,$d[2022] 210 4$d©2022 215 $a1 online resource (192 pages) 225 1 $aTutorials, schools, and workshops in the mathematical sciences 311 08$aPrint version: Ammari, Kaïs Research in PDEs and Related Fields Cham : Springer International Publishing AG,c2022 9783031142673 320 $aIncludes bibliographical references. 327 $aIntro -- Preface -- Contents -- Sobolev Spaces and Elliptic Boundary Value Problems -- 1 Sobolev Spaces, Inequalities, Dirichlet, and Neumann Problems for the Laplacian -- 1.1 Sobolev Spaces -- 1.2 First Properties -- 1.3 Traces -- 1.4 Interpolation -- 1.5 Transposition -- 1.6 Inequalities -- 1.7 Weak Solutions -- 1.8 Strong Solutions -- 1.9 Very Weak Solutions -- 1.10 Solutions in Hs(?), with 0 < -- s < -- 2 -- 2 The Stokes Problem with Various Boundary Conditions -- 2.1 The Problem (S) with Dirichlet Boundary Condition -- 2.2 The Stokes Problem with Navier Type Boundary Condition -- 2.3 The Stokes Problem with Navier Boundary Condition -- References -- Survey on the Decay of the Local Energy for the Solutions of the Nonlinear Wave Equation -- 1 Introduction and Preliminaries -- 2 Scattering for the Subcritical and Critical Wave Equation -- 2.1 The Subcritical Case -- 2.1.1 Prisized Morawetz Estimate -- 2.1.2 Global Time Strichartz Norms -- 2.1.3 The Proof of Theorem 2.1 -- 2.2 The Critical Case -- 2.2.1 Global Time Strichartz Norms -- 2.2.2 The Proof of Theorem 2.1 in the Case p=5 -- 3 Exponential Decay for the Local Energy of the Subcritical and Critical Wave Equation with Localized Semilinearity -- 3.1 Nonlinear Lax-Phillips Theory -- 3.2 Exponential Decay for the Local Energy of the Subcritical Wave Equation -- 3.2.1 The Compactness of Z(T) -- 3.2.2 Proof of Theorem 3.1 -- 3.3 Exponential Decay for the Local Energy of the Critical Wave Equation -- 4 Polynomial Decay for the Local Energy of the Semilinear Wave Equation with Small Data -- 4.1 Fundamental Lemmas -- 4.2 Proof of Theorem 4.1: Existence and Decay of the Local Energy -- 5 Decay of the Local Energy for the Solutions of the Critical Klein-Gordon Equation -- 5.1 Strichartz Norms Global in Time -- 5.2 Exponential Decay of the Local Energy of Localized Linear Klein-Gordon Equation. 327 $a5.2.1 Semi-Group of Lax-Phillips Adapted to Localized Linear Klein-Gordon Equation -- 5.2.2 Proof of Theorem 5.9 -- 5.3 Proof of Theorem 5.1 -- Appendix -- References -- A Spectral Numerical Method to Approximate the Boundary Controllability of the Wave Equation with Variable Coefficients -- 1 Introduction -- 2 Numerical Approximation of the Control Problem -- 3 Minimal L2-Weighted Controls -- 4 Numerical Experiments -- 5 Appendix -- References -- Aggregation Equation and Collapse to Singular Measure -- 1 Introduction -- 2 Graph Reformulation and Main Results -- 3 Dini and Hölder Spaces -- 4 Modified Curved Cauchy Operators -- 5 Local Well-Posedness -- 6 Global Well-Posedness -- 6.1 Weak and Strong Damping Behavior of the Source Term -- 6.2 Global a Priori Estimates -- References -- Geometric Control of Eigenfunctions of Schrödinger Operators -- 1 Introduction -- 2 The Geometric Control Condition -- 3 Are There Examples for Which (OE(?)) Holds and (OS(?)) Does Not? -- 4 A Geometric Interpretation of (V-GCC) and Proof of Theorem 9 -- 5 On the Proof of Theorem 10 -- References -- Stability of a Graph of Strings with Local Kelvin-Voigt Damping -- 1 Introduction -- 2 Well-Posedness of the System -- 3 Asymptotic Behavior -- References. 410 0$aTutorials, schools, and workshops in the mathematical sciences. 606 $aControl theory 606 $aDifferential equations, Partial 606 $aDifferential equations, Partial$xNumerical solutions 606 $aTeoria de control$2thub 606 $aEquacions en derivades parcials$2thub 606 $aSolucions numèriques$2thub 608 $aLlibres electrònics$2thub 615 0$aControl theory. 615 0$aDifferential equations, Partial. 615 0$aDifferential equations, Partial$xNumerical solutions. 615 7$aTeoria de control 615 7$aEquacions en derivades parcials 615 7$aSolucions numèriques 676 $a629.8312 702 $aAmmari$b Kai?s 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910624310303321 996 $aResearch in PDEs and related fields$93071095 997 $aUNINA