partial differential equation

- $n$Lab: differential equation, homological algebra in the finite element method, regular differential operator, variational calculus, diffiety, D-scheme, D-module, h-principle, pseudodifferential operator, polyfold
- MathOverflow: why-cant-there-be-a-general-theory-of-nonlinear-pde, recent-fundamental-new-directions-in-pdes, counterexamples-in-pde
- Sergiu Klainerman,
*PDE as a unified subject*, GAFA, Geom. Funct. Anal. pdf - Lars Hörmander,
*The analysis of linear partial differential operators*, 4 vols. Springer - J. J. Duistermaat,
*Fourier integral operators*, Progress in Math.**130**, Birkhauser 1996 - Gerald B. Folland,
*Introductions to PDEs*, Princeton Univ. Press 1995 - L. C. Evans,
*Partial differential equations*, Amer. Math. Soc. 1998 - David Gilbarg, Neil S. Trudinger,
*Elliptic differential equations of second order*, Springer 1977, 1983; rus. transl, Nauka 1989 - M. E. Taylor,
*Pseudodifferential operators*, Springer 1974 - M. Kashiwara, T. Kawai, T. Kimura,
*Foundations of algebraic analysis*, Princeton Univ. Press 1986 - M. Kashiwara,
*D-modules and microlocal analysis*, Transl. of math. monographs**217**, Amer. Math. Soc. 2000

- elliptic, hyperbolic and parabolic type equations
- globally those can mix, having different type in different regions (mixed PDEs)

- Cauchy-Kowalewski theorem (wikipedia) – quite general local existence and uniqueness theorem, but limited only to analytic coefficients and analytic solutions
- fixed point theorems (Banach, Schauder, Leray etc.): useful for evolution equations (including for ODEs, e.g. )
- index theorems
- asymptotic methods
- variational principles
- energies, conservation laws, dissipation
- various entropies
- a priori estimates
- global estimates
- Schauder estimates
- Harnack inequality
- h-principle
- characteristics of hyperbolic PDEs
- dispersion
- ellipticity
- transport
- incompressibility
- semigroups of operators

Great success of Fourier methods (esp. for constant coefficients).

Important classes:

- heat equation
- wave equation
- Helmholtz wave equation
- Laplace equation
- Maxwell equations
- self dual Maxwell equations
- Schroedinger equation
- Dirac equation
- Klein-Gordon equation
- Proca equation

- Einstein equations
- Euler equation
- Ricci flow
- Navier-Stokes equation
- Landau-Ginzburg equation
- Yang-Mills equations
- compressible Euler equation

- Monge-Ampere equation
- Cauchy-Riemann equation

- nonlinear Schroedinger equation (coming from quantum optics)
- Burgess equation
- KdV equation

- various types of weak solutions in spaces of functions and distributions
- (sheaves of) hyperfunctions
- D-modules
- parametrix
- fundamental solution and Green functions
- formal sections
- diffieties

important: propagation of singularities

- spaces of analytic functions
- $L_p$-spaces, Hoelder spaces, Sobolev spaces, Besov spaces
- distributions (and also densities and currents) of Sobolev, Schwarz, Coulombeau, hyperfunctions

Role of Fredholm theory, Sobolev and other embedding theorems, iterative schemes for improving regularity, interpolation spaces and so on.

- boundary value problems

- difference schemes
- finite element methods
- Monte-Carlo methods

- stochastic PDEs
- PDEs in cohesive topoi
- noncommutative PDEs
- difference equations
- integrodifferential equations
- differential equations for functions whose arguments are in Banach, Frechet spaces and so on
- second quantized equations in QFT
- fractional PDEs
- pseudodifferential operators and Fourier integral operators

Created on May 4, 2013 at 17:41:00. See the history of this page for a list of all contributions to it.