# Zoran Skoda partial differential equation

### Local behaviour

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

### Some fairly general theorems and principles

• 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

### Linear PDEs

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

### Important nonlinear PDEs

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

### Partial differential relations having large families of solutions

• Monge-Ampere equation
• Cauchy-Riemann equation
• nonlinear Schroedinger equation (coming from quantum optics)
• Burgess equation
• KdV equation

### Frameworks for defining solutions and defining their precursors

• 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

### Functional spaces

• 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.

### Types of problems

• boundary value problems

### Numerical methods

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

### Generalized PDEs and PDOs

• 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.