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1.

図書
図書
1. Progress in turbulence VII : proceedings of the iTi Conference in Turbulence 2016
Raims Örlü ... [et al.], editors
出版情報: Cham : Springer, c2017  xv, 250 p. ; 25 cm
シリーズ名: Springer proceedings in physics ; v. 196
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2.

図書
図書
2. Partial differential equations in fluid mechanics (: pbk)
edited by Charles L. Fefferman, James C. Robinson, José L. Rodrigo
出版情報: Cambridge : Cambridge University Press, 2018  ix, 326 p. ; 23 cm
シリーズ名: London Mathematical Society lecture note series ; 452
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3.

図書
図書
3. Geometrical properties of differential equations : applications of the Lie group analysis in financial mathematics
Ljudmila A. Bordag
出版情報: New Jersey : World Scientific, c2015  xi, 328 p. ; 24 cm
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4.

図書
図書
4. Partial differential equations. 2nd ed
Lawrence C. Evans
出版情報: Providence, R.I. : American Mathematical Society, c2010  xxi, 749 p. ; 27 cm
シリーズ名: Graduate studies in mathematics ; v. 19
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目次情報: 続きを見る
Introduction
Representation formulas for solutions: Four important linear partial differential equations
Nonlinear first-order PDE Other ways to represent solutions
Theory for linear partial differential equations: Sobolev spaces
Second-order elliptic equations Linear evolution equations
Theory for nonlinear partial differential equations: The calculus of variations
Nonvariational techniques Hamilton-Jacobi equations
Systems of conservation laws
Nonlinear wave equations
Appendices
Bibliography
Index
Introduction
Representation formulas for solutions: Four important linear partial differential equations
Nonlinear first-order PDE Other ways to represent solutions
5.

図書
図書
5. Glimpses of soliton theory : the algebra and geometry of nonlinear PDEs
Alex Kasman
出版情報: Providence, R.I. : American Mathematical Society, 2010  xvi, 304 p. ; 22 cm
シリーズ名: Student mathematical library ; v. 54
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6.

図書
図書
6. Mathematical physics with partial differential equations
James R. Kirkwood
出版情報: Amsterdam : Academic Press, c2013  xii, 418 p. ; 24 cm
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目次情報: 続きを見る
Preface
Preliminaries / 1：
Self-Adjoint Operators / 1-1：
Fourier Coefficients
Exercises
Curvilinear Coordinates / 1-2：
Scaling Factors
Volume Integrals
The Gradient
The Laplacian
Spherical Coordinates
Other Curvilinear Systems
Applications
An Alternate Approach (Optional)
Approximate Identities and the Dirac-δ Function / 1-3：
Approximate Identities
The Dirac-δ Function in Physics
Some Calculus for the Dirac-δ Function
The Dirac-δ Function in Curvilinear Coordinates
The Issue of Convergence / 1-4：
Series of Real Numbers
Convergence versus Absolute Convergence
Series of Functions
Power Series
Taylor Series
Some Important Integration Formulas / 1-5：
Other Facts We Will Use Later
Another Important Integral
Vector Calculus / 2：
Vector Integration / 2-1：
Path Integrals
Line Integrals
Surfaces
Parameterized Surfaces
Integrals of Scalar Functions Over Surfaces
Surface Integrals of Vector Functions
Divergence and Curl / 2-2：
Cartesian Coordinate Case
Cylindrical Coordinate Case
Spherical Coordinate Case
The Curl
The Curl in Cartesian Coordinates
The Curl in Cylindrical Coordinates
The Curl in Spherical Coordinates
Green's Theorem, the Divergence Theorem, and Stokes' Theorem / 2-3：
The Divergence (Gauss') Theorem
Stokes' Theorem
An Application of Stokes' Theorem
An Application of the Divergence Theorem
Conservative Fields
Green's Functions / 3：
Introduction
Construction of Green's Function Using the Dirac-δ Function / 3-1：
Construction of Green's Function Using Variation of Parameters / 3-2：
Construction of Green's Function from Eigenfunctions / 3-3：
More General Boundary Conditions / 3-4：
The Fredholm Alternative (or, What If 0 Is an Eigenvalue?) / 3-5：
Green's Function for the Laplacian in Higher Dimensions / 3-6：
Fourier Series / 4：
Basic Definitions / 4-1：
Methods of Convergence of Fourier Series / 4-2：
Fourier Series on Arbitrary Intervals
The Exponential Form of Fourier Series / 4-3：
Fourier Sine and Cosine Series / 4-4：
Double Fourier Series / 4-5：
Exercise
Three Important Equations / 5：
Laplace's Equation / 5-1：
Derivation of the Heat Equation in One Dimension / 5-2：
Derivation of the Wave Equation in One Dimension / 5-3：
An Explicit Solution of the Wave Equation / 5-4：
Converting Second-Order PDEs to Standard Form / 5-5：
Sturm-Liouville Theory / 6：
The Self-Adjoint Property of a Sturm-Liouville Equation / 6-1：
Completeness of Eigenfunctions for Sturm-Liouville Equations / 6-2：
Uniform Convergence of Fourier Series / 6-3：
Separation of Variables in Cartesian Coordinates / 7：
Solving Laplace's Equation on a Rectangle / 7-1：
Laplace's Equation on a Cube / 7-2：
Solving the Wave Equation in One Dimension by Separation of Variables / 7-3：
Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables / 7-4：
Solving the Heat Equation in One Dimension Using Separation of Variables / 7-5：
The Initial Condition Is the Dirac-δ Function
Steady State of the Heat Equation / 7-6：
Checking the Validity of the Solution / 7-7：
Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables / 8：
An Example Where Bessel Functions Arise
The Solution to Bessel's Equation in Cylindrical Coordinates / 8-1：
Solving Laplace's Equation in Cylindrical Coordinates Using Separation of Variables / 8-2：
The Wave Equation on a Disk (Drum Head Problem) / 8-3：
The Heat Equation on a Disk / 8-4：
Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables / 9：
An Example Where Legendre Equations Arise / 9-1：
The Solution to Bessel's Equation in Spherical Coordinates / 9-2：
Legendre's Equation and Its Solutions / 9-3：
Associated Legendre Functions / 9-4：
Laplace's Equation in Spherical Coordinates / 9-5：
The Fourier Transform / 10：
The Fourier Transform as a Decomposition / 10-1：
The Fourier Transform from the Fourier Series / 10-2：
Some Properties of the Fourier Transform / 10-3：
Solving Partial Differential Equations Using the Fourier Transform / 10-4：
The Spectrum of the Negative Laplacian in One Dimension / 10-5：
The Fourier Transform in Three Dimensions / 10-6：
The Laplace Transform / 11：
Properties of the Laplace Transform / 11-1：
Solving Differential Equations Using the Laplace Transform / 11-2：
Solving the Heat Equation Using the Laplace Transform / 11-3：
The Wave Equation and the Laplace Transform / 11-4：
Solving PDEs with Green's Functions / 12：
Solving the Heat Equation Using Green's Function / 12-1：
Green's Function for the Nonhomogeneous Heat Equation
The Method of Images / 12-2：
Method of Images for a Semi-infinite Interval
Method of Images for a Bounded Interval
Green's Function for the Wave Equation / 12-3：
Green's Function and Poisson's Equation / 12-4：
Appendix: Computing the Laplacian with the Chain Rule
References
Index
Preface
Preliminaries / 1：
Self-Adjoint Operators / 1-1：
7.

図書
図書
7. Fourier analysis and nonlinear partial differential equations (: hbk ; : pbk)
Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
出版情報: Heidelberg : Springer, c2011  xv, 523 p. ; 25 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; 343
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Basic Analysis / 1：
Basic Real Anslysis / 1.1：
Holder and Convolution Inequslities / 1.1.1：
The Atomic Decomposition / 1.1.2：
Proof of Refined Young Inequslityp8 / 1.1.3：
A Bilinear Interpolation Theorem / 1.1.4：
A Linear Interpolation Result / 1.1.5：
The Hardy-Littlewood Maximal Function / 1.1.6：
The Fourier Transform / 1.2：
Fourier Transforms of Functions and the Schwartz Space / 1.2.1：
Tempered Distributions and the Fourier Transform / 1.2.2：
A Few Calculations of Fourier Transforms / 1.2.3：
Homogeneous Sobolev Spaces / 1.3：
Definition and Basic Properties / 1.3.1：
Sobolev Embedding in Lebesgue Spaces / 1.3.2：
The Limit Case Hd/2 / 1.3.3：
The Embedding Theorem in Hölder Spaces / 1.3.4：
Nonhomogeneous Sobolev Spaces on Rd / 1.4：
Embedding / 1.4.1：
A Density Theorem / 1.4.3：
Hardy Inequality / 1.4.4：
References and Remarks / 1.5：
Littlewood-Paley Theory / 2：
Functions with Compactly Supported Fourier Transforms / 2.1：
Bernstein-Type Lemmas / 2.1.1：
The Smoothing Effect of Heat Flow / 2.1.2：
The Action of a Diffeomorphism / 2.1.3：
The Effects of Some Nonlinear Functions / 2.1.4：
Dyadic Partition of Unity / 2.2：
Homogeneous Besov Spaces / 2.3：
Characterizations of Homogeneous Besov Spaces / 2.4：
Besov Spaces, Lebesgue Spaces, and Refined Inequalities / 2.5：
Homogeneous Paradifferential Calculus / 2.6：
Homogeneous Bony Decomposition / 2.6.1：
Action of Smooth Functions / 2.6.2：
Time-Space Besov Spaces / 2.6.3：
Nonhomogeneous Besov Spaces / 2.7：
Nonhomogeneous Paradifferential Calculus / 2.8：
The Bony Decomposition / 2.8.1：
The Paralinearization Theorem / 2.8.2：
Besov Spaces and Compact Embeddings / 2.9：
Commutator Estimates / 2.10：
Around the Space B&infty;,&infty;1 / 2.11：
Transport and Transport-Diffusion Equations / 2.12：
Ordinary Differential Equations / 3.1：
The Cauchy-Lipschitz Theorem Revisited / 3.1.1：
Estimates for the Flow / 3.1.2：
A Blow-up Criterion for Ordinary Differential Equations / 3.1.3：
Transport Equations: The Lipschitz Case / 3.2：
A Priori Estimates in General Besov Spaces / 3.2.1：
Refined Estimates in Besov Spaces with Index 0 / 3.2.2：
Solving the Transport Equation in Besov Spaces / 3.2.3：
Application to a Shallow Water Equation / 3.2.4：
Losing Estimates for Transport Equations / 3.3：
Linear Loss of Regularity in Besov Spaces / 3.3.1：
The Exponential Loss / 3.3.2：
Limited Loss of Regularity / 3.3.3：
A Few Applications / 3.3.4：
Transport-Diffusion Equations / 3.4：
A Priori Estimates / 3.4.1：
Exponential Decay / 3.4.2：
Quasilinear Symmetric Systems / 3.5：
Definition and Examples / 4.1：
Linear Symmetric Systems / 4.2：
The Well-posedness of Linear Symmetric Systems / 4.2.1：
Finite Propagation Speed / 4.2.2：
Further Well-posedness Results for Linear Symmetric Systems / 4.2.3：
The Resolution of Quasilinear Symmetric Systems / 4.3：
Paralinearization and Energy Estimates / 4.3.1：
Convergence of the Scheme / 4.3.2：
Completion of the Proof of Existence / 4.3.3：
Uniqueness and Continuation Criterion / 4.3.4：
Data with Critical Regularity and Blow-up Criteria / 4.4：
Critical Besov Regularity / 4.4.1：
A Refined Blow-up Crndition / 4.4.2：
Continuity of the Flow Map / 4.5：
The Incompressible Navier-Stokes System / 4.6：
Basic Facts Concerning the Navier-Stokes System / 5.1：
Well-posedness in Sobolev Spaces / 5.2：
A General Result / 5.2.1：
The Behavior of the Hd/2-1 Norm Near 0 / 5.2.2：
Results Related to the Structure of the System / 5.3：
The Particular Case of Dimension Two / 5.3.1：
The Case of Dimension Three / 5.3.2：
An Elementary Lp Approach / 5.4：
The Endpoint Space for Picard's Scheme / 5.5：
The Use of the L1-smoothing Effect of the Heat Flow / 5.6：
The Cannone-Meyer-Planchon Theorem Revisited / 5.6.1：
The Flow of the Solutions of the Navier-Stokes System / 5.6.2：
Anisotropic Viscosity / 5.7：
The Case of L2 Data with One Vertical Derivative in L2 / 6.1：
A Global Existence Result in Anisotropic Besov Spaces / 6.2：
Anisotropic Localization in Fourier Space / 6.2.1：
The Functional Framework / 6.2.2：
Statement of the Main Result / 6.2.3：
Some Technical Lemmas / 6.2.4：
The Proof of Existence / 6.3：
The Proof of Uniqueness / 6.4：
Euler System for Perfect Incompressible Fluids / 6.5：
Local Well-posedness Results for Inviscid Fluids / 7.1：
The Biot-Savart Law / 7.1.1：
Estimates for the Pressure / 7.1.2：
Another Formulation of the Euler System / 7.1.3：
Local Existence of Smooth Solutions / 7.1.4：
Uniqueness / 7.1.5：
Continuation Criteria / 7.1.6：
Global Existence Results in Dimension Two / 7.2：
Smooth Solutions / 7.2.1：
The Borderline Case / 7.2.2：
The Yudovich Theorem / 7.2.3：
The Inviscid Limit / 7.3：
Regularity Results for the Navier-Stokes System / 7.3.1：
The Smooth Case / 7.3.2：
The Rough Case / 7.3.3：
Viscous Vortex Patches / 7.4：
Results Related to Striated Regularity / 7.4.1：
A Stationary Estimate for the Velocity Field / 7.4.2：
Uniform Estimates for Striated Regularity / 7.4.3：
A Global Convergence Result for Striated Regularity / 7.4.4：
Application to Smooth Vortex Patches / 7.4.5：
Strichartz Estimates and Applications to Semilinear Dispersive Equations / 7.5：
Examples of Dispersive Estimates / 8.1：
The Dispersive Estimate for the Free Transport Equation / 8.1.1：
The Dispersive Estimates for the Schrdillger Equation / 8.1.2：
Integral of Oscillating Functions / 8.1.3：
Dispersive Estimates for the Wave Equation / 8.1.4：
The L2 Boundedness of Some Fourier Integral Operators / 8.1.5：
Billnear Methods / 8.2：
The Duality Method and the TT* Argument / 8.2.1：
Strichartz Estimates: The Case q > 2 / 8.2.2：
Strichartz Estimates: The Endpoint Case q = 2 / 8.2.3：
Application to the Cubic Semilinear Schrödinger Equation / 8.2.4：
Strichartz Estimates for the Wave Equation / 8.3：
The Basic Strichartz Estimate / 8.3.1：
The Refined Strichartz Estimate / 8.3.2：
The Qulntic Wave Equation in R3 / 8.4：
The Cubic Wave Equation in R3 / 8.5：
Solutions in H1 / 8.5.1：
Local and Global Well-posedness for Rough Data / 8.5.2：
The Nonlinear Interpolation Method / 8.5.3：
Application to a Class of Semilinear Wave Equations / 8.6：
Smoothing Effect in Quasilinear Wave Equations / 8.7：
A Well-posedness Result Based on an Energy Method / 9.1：
The Main Statement and the Strategy of its Proof / 9.2：
Refined Paralinearization of the Wave Equation / 9.3：
Reduction to a Microlocal Strichartz Estimate / 9.4：
Microlocal Strichartz Estimates / 9.5：
A Rather General Statement / 9.5.1：
Geometrical Optics / 9.5.2：
The Solution of the Eikonal Equation / 9.5.3：
The Transport Equation / 9.5.4：
The Approximation Theorem / 9.5.5：
The Proof of Theorem 9.16 / 9.5.6：
The Compressible Navier-Stokes System / 9.6：
About the Model / 10.1：
General Overview / 10.1.1：
The Barotropic Navier-Stokes Equations / 10.1.2：
Local Theory for Data with Critical Regularity / 10.2：
Scaling Invariance and Statement of the Main Result / 10.2.1：
Existence of a Local Solution / 10.2.2：
A Continuation Criterion / 10.2.4：
Local Theory for Data Bounded Away from the Vacuum / 10.3：
A Priori Estimates for the Linearized Momentum Equation / 10.3.1：
Global Existence for Small Data / 10.3.2：
Statement of the Results / 10.4.1：
A Spectral Analysis of the Linearized Equation / 10.4.2：
A Prioli Estimates for the Linearized Equation / 10.4.3：
Proof of Global Existence / 10.4.4：
The Incompressible Limit / 10.5：
Main Results / 10.5.1：
The Case of Small Data with Critical Regularity / 10.5.2：
The Case of Large Data with More Regularity / 10.5.3：
References / 10.6：
List of Notations
Index
Basic Analysis / 1：
Basic Real Anslysis / 1.1：
Holder and Convolution Inequslities / 1.1.1：
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