Selected Models of Transport Processes. Methods of Solving and Properties of Solutions (2010)

Nonlinear transport phenomena: models, method of solving and unusual features

1. Heat transport equation.
   1. Thermal explosion, linear case. A concept of self-similar solution.
   2. Thermal explosion in nonlinear cases. Finite velocity of perturbations and localization of heat energy.
   3. Boundary value problem: effect of the complete heat localization. A concept of blow-up regimes.
   4. Asymptotic properties of the solutions of the problem of thermal explosion.
   5. Maximum principle, comparison theorems and their consequences.
2. Burgers equation and related models.
   1. Burgers equation (BE) and its hyperbolic generalization (GBE).
   2. Cole-Hopf transformation. Solution of the Cauchý problem to BE.
   3. Asypmtotic properties. Dependence on the “Reynolds number” R.
   4. GBE: appearance of the “Mach number” M. Overview of the cases M >1 and M <1.
   5. GBE with sources: analytical description of the traveling wave solutions.
3. Korteveg de Vries (KdV) equation.
   1. Historical background: observation of Scott Russel. The first discovery of KdV equation and soliton’s analytical description. The second half of the XX century: rediscovery of solitons.
   2. Two solitons’ interaction: description based upon the Hirota method, and results of numerical simulation. Nonlinear analog of the superposition principle.
   3. Existence of “soliton domain” in the set of smooth Cauchý data. Multisoliton solutions.
   4. Bäclund transformation. A concept of complete integrability of a nonlinear PDE.
   5. KdV equation: and infinite hierarchy of conservation laws.
   6. Localized wave patterns (solitary waves, compactons, shock fronts and all that) within the models, which are not completely integrable.

Language of the course - English.

Tutor of the course:

Prof. V. Vladimirov, AGH University of Science and Technology, Poland