Abstract
We construct a uniform in time asymptotics describing the interaction of two isothermal shock waves with opposite directions of motion. We show that any smooth regularization of the problem implies the realization of the stable scenario of interaction.
Highlights
We consider the gas dynamics system in the isothermal case ∂ρ ∂t + ∂ ∂x =0, x ∈ R1, t > 0, ∂ ∂t ∂ ∂x ρu2 + c02ρ = 0, (1.1)
At the time t∗, the initial conditions (1.2) are replaced by the shock wave with the amplitudes ρ1 − ρ2 and u1 − u2 of the jumps of ρ and u, which are concentrated at the point x = x∗
Solving this Riemann problem, we obtain that the solution for t > t∗ is again represented by two noninteracting shock waves with uniquely defined new amplitudes and new paths of propagation
Summary
At the time t∗, the initial conditions (1.2) are replaced by the shock wave with the amplitudes ρ1 − ρ2 and u1 − u2 of the jumps of ρ and u, which are concentrated at the point x = x∗ Solving this Riemann problem, we obtain that the solution for t > t∗ is again represented by two noninteracting shock waves with uniquely defined new amplitudes and new paths of propagation (see, e.g., [2, 9]). Whitham found the exact solution for the initial data similar to (1.2) and, as a result, established that the regularization implies the choice of a stable scenario of interaction This procedure works uniquely for the quadratic nonlinearity. This can be done, and we obtain a uniform in time description of the interaction of two shock waves in the case of opposite directions of motion
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