Описание:This self-contained course is devoted to the presentation of classical and recent results of the theory of nonlinear partial differential equations. First of all, we consider first order quasi-linear equations and systems of conservation laws, in particular, the systems which are used to model many continuous media. We discuss the issues of singularity formation, the construction of shock wave solutions and their regularization, large time asymptotics.
Then we consider equations of higher orders which model important processes in physics and biology. This is the Kolmogorov-Petrovskii-Piskunov equation, Korteweg-de Vries equation, the nonlinear Schrödinger equation, nonlinear heat conduction equations. We discuss the traveling wave solutions, formation and propagation of singularities, solitons, the localization of heat fronts. The course aims to familiarize with the most striking phenomena of the theory of nonlinear equations.
1. Equations of the first order. Conservation laws. Linear, quasi-linear and non-linear equations. Hamilton-Jacobi equations. The characteristics and their role. The characteristic system of equations.
2. Formation of singularities of a smooth solution of the quasilinear equation.
3. The generalized solution. Rankine-Hugoniot conditions. The problem of uniqueness . Admissibility.
4. The method of vanishing viscosity for a quasi-linear equation. The Burgers equation.
5. The Cauchy problem in the class of admissible solutions. The proof of uniqueness.
6. The existence of an admissible solution of the Cauchy problem for a quasi-linear equation. The Lax-Oleinik formula.
7. Large time behavior of the admissible solution. Asymptotics in different norms. Decay into N-wave.
8. Hyperbolic systems of conservation laws. Genuinely nonlinear and linear degenerate characteristic directions. Admissibility of shicks according to Lax. Small shocks. K-waves: shocks, rarefaction waves, contact discontinuities.
9. Entropy criteria. Construction of entropy for an equation and a system of two equations.
10. Symmetric hyperbolic systems and entropy.
11. Examples of hyperbolic systems in a strict and non-strict sense. The system of equations of gas dynamics. Pressureless gas dynamics. Delta-shocks.
12. Riemann invariants. The symmetric form of a hyperbolic system. The system of gas dynamics equations in invariants.
13. The Kolmogorov-Petrovskii-Piskunov equation. Existence of a solution in the form of a traveling wave. The stability of this solution.
14. Various methods of regularization the Hopf equation: the viscocity and the dispersion. Self-similar solution. The relation between the orders of viscosity and dispersion.
15. The Korteweg-de Vries equation. The solution in the form of a solitary wave. Solitons. Construction of a two-soliton solution.
16. Nonlinear Schrödinger equation. The concept of a critical nonlinearity. The method of moments. Formation of singularity. Hydrodynamic interpretation.
17. Non-linear heat equation. Heat wave propagation: finite and infinite speed of the wave, the smoothness of the solution at the front, the phenomenon of localization of heat.
18. Integral functionals ( generalized moments) for the gas dynamics like systems. Application to the singularity formation. Special classes of solutions. Reduction to ODE systems. Applications to the vortex dynamics.
19. Method of stochastic regularization for quasi-linear equations. Correspondence between stochastic and viscous regularizations. From motion of particles in physics to portfolio selection in finance.