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Evolution of two-dimensional contrast structures of complex form. (English. Russian original) Zbl 0965.35124
Comput. Math. Math. Phys. 39, No. 5, 769-778 (1999); translation from Zh. Vychisl. Mat. Mat. Fiz. 39, No. 5, 801-811 (1999).
Summary: Formation and decay of unsteady two-dimensional contrast structures, i.e., of a certain class of solutions of a two-dimensional nonlinear diffusion equation with convection and generation, are analyzed. The contrast structures form as a result of the saturation of a solution in absolute value at a certain level and are characterized by the existence of relatively large domains (spots) in which the solution is close in absolute value to the saturation level and has a small gradient. These domains are separated by relatively narrow transition layers in which the field reverses its sign and has a steep gradient. The transition layers slowly move, and the contrast structure eventually decays. The drift velocity of the curved boundary of a contrast structure is calculated, and a formula for the lifetime of an arbitrary contrast structure is obtained. The drift velocity of a transition layer of prescribed form is calculated, and approximate formulas describing the evolution of contrast-structure spots of circular, elliptic, and other simple forms are obtained. The results of a numerical experiment are presented for the two-dimensional time-dependent model.
35Q35 PDEs in connection with fluid mechanics
76R50 Diffusion
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs