Nonlinear Partial Differential Equations : Asymptotic Behavior of Solutions and Self-Similar Solutions


Mi-Ho. Giga
Bok Engelsk 2010 · Electronic books.
Annen tittel
Medvirkende
Utgitt
Boston, Mass. : Birkhäuser , c2010
Omfang
1 online resource (306 p.)
Opplysninger
Description based upon print version of record.. - ""Nonlinear Partial Differential Equations ""; ""Contents ""; ""Preface""; ""Part I Asymptotic Behavior of Solutions of Partial Differential Equations""; ""1 Behavior Near Time Infinity of Solutions of the Heat Equation""; ""1.1 Asymptotic Behavior of Solutions Near Time Infinity""; ""1.1.1 Decay Estimate of Solutions""; ""1.1.2 Lp-Lq Estimates""; ""1.1.3 Derivative Lp-Lq Estimates""; ""1.1.4 Theorem on Asymptotic Behavior Near Time Infinity""; ""1.1.5 Proof Using Representation Formula of Solutions""; ""1.1.6 Integral Form of the Mean Value Theorem"". - ""1.2 Structure of Equations and Self-Similar Solutions""""1.2.1 Invariance Under Scaling""; ""1.2.2 Conserved Quantity for the Heat Equation""; ""1.2.3 Scaling Transformation Preserving the Conserved Quantity""; ""1.2.4 Summary of Properties of a Scaling Transformation""; ""1.2.5 Self-Similar Solutions""; ""1.2.6 Expression of Asymptotic Formula Using Scaling Transformations""; ""1.2.7 Idea of the Proof Based on Scaling Transformation""; ""1.3 Compactness""; ""1.3.1 Family of Functions Consisting of Continuous Functions""; ""1.3.2 Ascoliâ€?Arzelà-type Compactness Theorem"". - ""1.3.3 Relative Compactness of a Family of Scaled Functions""""1.3.4 Decay Estimates in Space Variables""; ""1.3.5 Existence of Convergent Subsequences""; ""1.3.6 Lemma""; ""1.4 Characterization of Limit Functions""; ""1.4.1 Limit of the Initial Data""; ""1.4.2 Weak Form of the Initial Value Problem for the Heat Equation""; ""1.4.3 Weak Solutions for the Initial Value Problem""; ""1.4.4 Limit of a Sequence of Solutions to the Heat Equation""; ""1.4.5 Characterization of the Limit of a Family of Scaled Functions""; ""1.4.6 Uniqueness Theorem When Initial Data is the Delta Function"". - ""1.4.7 Completion of the Proof of Asymptotic Formula (1.9) Based on Scaling Transformation""""1.4.8 Remark on Uniqueness Theorem""; ""2 Behavior Near Time Infinity of Solutions of the Vorticity Equations""; ""2.1 Navierâ€?Stokes Equations and Vorticity Equations""; ""2.1.1 Vorticity""; ""2.1.2 Vorticity and Velocity""; ""2.1.3 Biotâ€?Savart Law""; ""2.1.4 Derivation of the Vorticity Equations""; ""2.2 Asymptotic Behavior Near Time Infinity""; ""2.2.1 Unique Existence Theorem""; ""2.2.2 Theorem for Asymptotic Behavior of the Vorticity""; ""2.2.3 Scaling Invariance"". - ""2.2.4 Conservation of the Total Circulation""""2.2.5 Rotationally Symmetric Self-Similar Solutions""; ""2.3 Global Lq-L1 Estimates for Solutions of the Heat Equation with a Transport Term""; ""2.3.1 Fundamental Lq-Lr Estimates""; ""2.3.2 Change Ratio of Lr-Norm per Time: Integral Identities""; ""2.3.3 Nonincrease of L1-Norm""; ""2.3.4 Application of the Nash Inequality""; ""2.3.5 Proof of Fundamental Lq-L1 Estimates""; ""2.3.6 Extension of Fundamental Lq-L1 Estimates""; ""2.3.7 Maximum Principle""; ""2.3.8 Preservation of Nonnegativity"". - ""2.4 Estimates for Solutions of Vorticity Equations"". - The main focus of this textbook, in two parts, is on showing how self-similar solutions are useful in studying the behavior of solutions of nonlinear partial differential equations, especially those of parabolic type. The exposition moves systematically from the basic to more sophisticated concepts with recent developments and several open problems. With challenging exercises, examples, and illustrations to help explain the rigorous analytic basis for the Navier–-Stokes equations, mean curvature flow equations, and other important equations describing real phenomena, this book is written for graduate students and researchers, not only in mathematics but also in other disciplines. Nonlinear Partial Differential Equations will serve as an excellent textbook for a first course in modern analysis or as a useful self-study guide. Key topics in nonlinear partial differential equations as well as several fundamental tools and methods are presented. The only prerequisite required is a basic course in calculus.
Emner
Sjanger
Dewey
ISBN
9780817641733

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