Mathematical Theory of Compressible Viscous Fluids: Analysis and Numerics (Advances in Mathematical Fluid Mechanics)

This book offers an essential introduction to the mathematical theory of compressible viscous fluids. The main goal is to present analytical methods from the perspective of their numerical applications. Accordingly, we introduce the principal theoretical tools needed to handle well-posedness of the underlying Navier-Stokes system, study the problems of sequential stability, and, lastly, construct solutions by means of an implicit numerical scheme. Offering a unique contribution -- by exploring in detail the "synergy" of analytical and numerical methods -- the book offers a valuable resource for graduate students in mathematics and researchers working in mathematical fluid mechanics. Mathematical fluid mechanics concerns problems that are closely connected to real-world applications and is also an important part of the theory of partial differential equations and numerical analysis in general. This book highlights the fact that numerical and mathematical analysis are not two separate fields of mathematics. It will help graduate students and researchers to not only better understand problems in mathematical compressible fluid mechanics but also to learn something from the field of mathematical and numerical analysis and to see the connections between the two worlds. Potential readers should possess a good command of the basic tools of functional analysis and partial differential equations including the function spaces of Sobolev type "This book offers an essential introduction to the mathematical theory of compressible viscous fluids. The main goal is to present analytical methods from the perspective of their numerical applications. Accordingly, we introduce the principal theoretical tools needed to handle well-posedness of the underlying Navier-Stokes system, study the problems of sequential stability, and, lastly, construct solutions by means of an implicit numerical scheme. Offering a unique contribution--by exploring in detail the "synergy" of analytical and numerical methods--the book offers a valuable resource for graduate students in mathematics and researchers working in mathematical fluid mechanics. Mathematical fluid mechanics concerns problems that are closely connected to real-world applications and is also an important part of the theory of partial differential equations and numerical analysis in general. This book highlights the fact that numerical and mathematical analysis are not two separate fields of mathematics. It will help graduate students and researchers to not only better understand problems in mathematical compressible fluid mechanics but also to learn something from the field of mathematical and numerical analysis and to see the connections between the two worlds. Potential readers should possess a good command of the basic tools of functional analysis and partial differential equations including the function spaces of Sobolev type."--Back cover Front Matter....Pages i-xii Preliminaries, Notation, and Spaces of Functions....Pages 1-22 Front Matter....Pages 23-23 Mathematical Model....Pages 25-30 Weak Solutions....Pages 31-38 A Priori Bounds....Pages 39-47 Weak Formulation Revisited....Pages 49-53 Weak Sequential Stability....Pages 55-77 Front Matter....Pages 79-79 Numerical Method....Pages 81-102 Stability of the Numerical Method....Pages 103-109 Consistency....Pages 111-133 Convergence....Pages 135-154 Front Matter....Pages 155-155 Weak Solutions with Artificial Pressure....Pages 157-164 Strong Convergence of the Approximate Densities....Pages 165-176 Concluding Remarks and Suggestions for Further Reading....Pages 177-178 Back Matter....Pages 179-186