Steady and Self-Similar Solutions to Two-Dimensional Hyperbolic Conservation Laws.

[摘要] We consider steady and self-similar solutions to two-dimensional systems of conservation laws that are small (in uniform norm) perturbations of a constant background solution.Assuming that the characteristic fields for the steady problem are of constant multiplicity, and that the simple eigenvalues are either genuinely nonlinear or linearly degenerate, we are able to prove that admissible solutions are special functions of variation.In addition, any such solution must be constant outside of thin sectors centered at the characteristic directions of the background state, and we characterize whether the behavior in each genuinely nonlinear sector is analogous to a forward or backward in time solution to a system of conservation laws in one spatial dimension.Each sector corresponding to a linearly degenerate field can contain at most a single contact discontinuity, and forward genuinely nonlinear sectors can contain at most a single shock or simple wave.Backward genuinely nonlinear sectors can contain infinitely many shocks and simple waves, though two consecutive simple waves cannot occur.Physical one-dimensional conservation laws with eigenvalues of constant multiplicity and either linearly degenerate or genuinely nonlinear simple eigenvalues satisfy these assumptions, as does steady isentropic and full two-dimensional Euler flow, provided the background state is supersonic.Steady full Euler flow also satisfies the hypotheses for supersonic background states.We also consider steady and self-similar solutions to the two-dimensional full Euler equations.Instead of assuming they are small perturbations of a constant supersonic background, we instead assume they are bounded with density and internal energy bounded away from zero, and nonvanishing velocity.Then, we show such a solution must be a special function of bounded variation, and we also obtain some results concerning the possible structure of these solutions.

[发布机构] University of Michigan

[效力级别] Riemann Problem

[关键词] Conservation Law;Riemann Problem;Bounded Variation;Self-similar;Compressible Euler;Mathematics;Science;Mathematics