Recent Advances in PDEs: Analysis, Numerics and Control: In Honor of Prof. Fernández-Cara's 60th Birthday (SEMA SIMAI Springer Series, 17)

This book contains the main results of the talks given at the workshop “Recent Advances in PDEs: Analysis, Numerics and Control”, which took place in Sevilla (Spain) on January 25-27, 2017. The work comprises 12 contributions given by high-level researchers in the partial differential equation (PDE) area to celebrate the 60th anniversary of Enrique Fernández-Cara (University of Sevilla). The main topics covered here are: Control and inverse problems, Analysis of Fluid mechanics and Numerical Analysis. The work is devoted to researchers in these fields. Introduction 6 Foreword 7 Contents 11 Short Biography 13 Essential Spectrum and Null Controllability of Some ParabolicEquations 14 1 Introduction and Main Results 14 2 Essential Spectrum, Singular Sequences and Controllability 16 3 The Controllability Issue for Degenerate Parabolic Equations 19 3.1 Proof of Theorem 1 19 3.2 Comments 22 4 The Controllability Issue for Mixed Parabolic Systems 23 4.1 Proof of Theorem 2 23 4.2 Comments 26 References 27 Well-Posedness and Asymptotic Behavior for a Nonlinear WaveEquation 29 1 Introduction 29 2 Some Notations and Results 31 3 Well-Posedness 33 4 Asymptotic Behavior 41 References 43 A Second-Order Linear Newmark Method for Lagrangian Navier-Stokes Equations 45 1 Introduction 46 2 Statement of Problem in Eulerian Coordinates and Notations 48 3 Strong Problem and Weak Formulation in Lagrangian Coordinates 50 4 Time Discretization: Linear Newmark Characteristic Method 52 5 Space Discretization: Finite Element Method 57 6 Numerical Results 61 References 69 Lubrication Theory and Viscous Shallow-Water Equations 72 1 Introduction 73 2 Derivation from Shallow-Water Equations 75 3 Relative Entropy and Strong Convergences 78 4 Change of Time by Y. Brenier and X. Duan 80 References 81 The Influence of the Tikhonov Term in Optimal Control of Partial Differential Equations 83 1 Introduction 83 2 About the Existence and Uniqueness of Optimal Controls 87 2.1 Existence of Solution of (P) 88 2.2 About the Uniqueness of Solution of (P) 89 3 Regularity of Optimal Solutions of State-Constrained Control Problems 91 4 Convergence of the Numerical Approximations 96 5 Second Order Analysis 97 5.1 Case λ> 0 101 5.2 Case λ= 0 103 References 104 On the Design of Algebraic Fractional Step Methods for Viscoelastic Incompressible Flows 105 1 Introduction 106 2 Problem Statement and Numerical Approximation 107 3 Schemes Based on Pressure Extrapolation 110 3.1 Formulation of the Algorithms 110 3.2 Equivalent Monolithic Formulations 113 4 Schemes Based on Velocity Extrapolation 115 4.1 The Continuous Problem 115 4.2 Formulation of the Algorithms 116 4.3 Equivalent Monolithic Formulation 117 5 Comments on Stability 118 6 The Inexact Factorization Point of View 120 6.1 First Order Pressure Extrapolation Scheme as Inexact Factorization 120 6.2 First Order Velocity Extrapolation Scheme as Inexact Factorization 121 7 Conclusions 123 References 124 Some Remarks on the Hierarchic Control for CoupledParabolic PDEs 126 1 Introduction 126 2 The Problem and Its Formulation 127 2.1 Main Results 130 3 Proof of Theorem 1 132 3.1 Optimality Condition for the Followers 132 3.2 The Leader Controls 133 3.3 Proof of the Observability Inequality 135 4 Proof of Theorem 2 141 5 Concluding Remarks 144 References 145 Local Null Controllability of the N-Dimensional Ladyzhenskaya-Smagorinsky with N-1 Scalar Controls 147 1 Introduction and Main Results 147 2 Some Technical Results 150 2.1 Carleman Estimates 150 2.2 Null Controllability of (1.3) and (1.4) 153 2.3 Estimates for the States Solutions 154 3 The Proof of Theorem 1.1 158 4 The Proof of Theorem 1.2 163 5 Some Open Problems 165 References 166 Numerical Estimations of the Cost of Boundary Controls for the Equation yt- yxx+M yx=0 with Respect to 167 1 Introduction: Problem Statement 167 2 Reformulation of the Controllability Cost K(,T,M) 169 3 Approximation of the Control Problem 173 3.1 Mixed Variational Formulation 174 3.1.1 Mixed Formulation 174 3.1.2 Minimization with Respect to the Lagrange Multiplier 177 3.2 Numerical Approximation 178 3.2.1 The Discrete inf-sup Test 179 3.3 Numerical Experiments 181 4 Numerical Approximation of the Cost of Control 188 4.1 Cost of Control in the Case M=1 189 4.2 Controllability Cost in the Case M=-1 192 5 Concluding Remarks and Perspectives 194 Appendix 197 References 198 Control of Random PDEs: An Overview 200 1 Introduction: Some Motivating Examples 200 1.1 Uncertainty Is Almost Everywhere 201 1.2 How to Model Uncertainty in PDEs-Based Models? 201 2 Existence of Solutions for Random PDEs and for Its Associated Optimal Control Problems 205 2.1 Variational Formulation of Random PDEs 205 2.2 Existence of Solutions for Robust and Risk Averse Control Problems 208 3 Numerical Resolution of Robust and Risk Averse Optimal Control Problems 209 3.1 Numerical Approximation of Robust Optimal Control Problems 209 3.2 Numerical Approximation of Risk Averse Control Problems 212 4 Further Comments and Challenging Problems 215 References 216 The Dubovitskii and Milyutin Formalism Applied to an Optimal Control Problem in a Solidification Model 218 1 Introduction 219 2 Preliminaries 224 2.1 Embeddings in Time-Spatial Dependent Sobolev Spaces 224 2.2 Results of the Solidification Model (1) 224 2.3 The Dubovitskii and Milyutin Formalism 225 2.4 Some Concepts of Differential Calculus 227 2.5 Some Generic Results for Explicit Calculus of Cones 228 3 Reformulation of the Optimal Control Problem and Specific Calculus of Cones 228 4 Proof of Theorem 1 231 5 Proof of Theorem 2 232 6 Proof of Theorem 3 234 6.1 Existence 234 6.2 Uniqueness 237 References 237 Local Regularity for Fractional Heat Equations 239 1 Introduction 240 2 Regularity Results for the Elliptic Problem 244 3 Proof of Theorem 1.5 250 3.1 The W2s,2-Regularity on RN 250 3.2 The W2s,2loc-Regularity in Ω 251 4 Proof of Theorem 1.6 252 5 Open Problems and Perspectives 254 References 255 Front Matter ....Pages i-xiii Essential Spectrum and Null Controllability of Some Parabolic Equations (Farid Ammar Khodja, Cédric Dupaix)....Pages 1-15 Well-Posedness and Asymptotic Behavior for a Nonlinear Wave Equation (Fágner Dias Araruna, Frederico de Oliveira Matias, Milton de Lacerda Oliveira, Shirley Maria Santos e Souza)....Pages 17-32 A Second-Order Linear Newmark Method for Lagrangian Navier-Stokes Equations (Marta Benítez, Alfredo Bermúdez, Pedro Fontán)....Pages 33-59 Lubrication Theory and Viscous Shallow-Water Equations (Didier Bresch, Mathieu Colin, Xi Lin, Pascal Noble)....Pages 61-71 The Influence of the Tikhonov Term in Optimal Control of Partial Differential Equations (Eduardo Casas)....Pages 73-94 On the Design of Algebraic Fractional Step Methods for Viscoelastic Incompressible Flows (Ramon Codina)....Pages 95-115 Some Remarks on the Hierarchic Control for Coupled Parabolic PDEs (Víctor Hernández-Santamaría, Luz de Teresa)....Pages 117-137 Local Null Controllability of the N-Dimensional Ladyzhenskaya-Smagorinsky with N-1 Scalar Controls (Dany Nina Huaman, Juan Límaco, Miguel R. Nuñez Chávez)....Pages 139-158 Numerical Estimations of the Cost of Boundary Controls for the Equation yt − εyxx + Myx = 0 with Respect to ε (Arnaud Münch)....Pages 159-191 Control of Random PDEs: An Overview (Francisco J. Marín, Jesús Martínez-Frutos, Francisco Periago)....Pages 193-210 The Dubovitskii and Milyutin Formalism Applied to an Optimal Control Problem in a Solidification Model (Aníbal Coronel, Francisco Guillén-González, Francisco Marques-Lopes, Marko Rojas-Medar)....Pages 211-231 Local Regularity for Fractional Heat Equations (Umberto Biccari, Mahamadi Warma, Enrique Zuazua)....Pages 233-249 This book contains the main results of the talks given at the workshop "Recent Advances in PDEs: Analysis, Numerics and Control", which took place in Sevilla (Spain) on January 25-27, 2017. The work comprises 12 contributions given by high-level researchers in the partial differential equation (PDE) area to celebrate the 60th anniversary of Enrique Fernández-Cara (University of Sevilla). The main topics covered here are: Control and inverse problems, Analysis of Fluid mechanics and Numerical Analysis. The work is devoted to researchers in these fields.-- Provided by publisher