Global entropy solutions to multi-dimensional isentropic gas dynamics with spherical symmetry

Abstract

We are concerned with spherically symmetric solutions to the Euler equations for the multi-dimensional compressible fluids, which have many applications in diverse real physical situations. The system can be reduced to one dimensional isentropic gas dynamics with geometric source terms. Due to the presence of the singularity at the origin, there are few papers devoted to this problem. The present paper proves two existence theorems of global entropy solutions. The first one focuses on the case excluding the origin in which the negative velocity is allowed, and the second one is corresponding to the case including the origin with non-negative velocity. The L ∞ compensated compactness framework and vanishing viscosity method are applied to prove the convergence of approximate solutions. In the second case, we show that if the blast wave initially moves outwards and the initial densities and velocities decay to zero with certain rates near origin, then the densities and velocities tend to zero with the same rates near the origin for any positive time. In particular, the entropy solutions in two existence theorems are uniformly bounded with respect to time.

Large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductor devices

“…Therefore, Lemma 3.2 is completed. From (3.2), the velocity u = J n is uniformly bounded, i.e., |u| ≤ C. Then following the same way of [15], we can obtain n(x, t) ≥ δ(t, ε) > 0, (3.24) where δ(t, ε) depends on n 0 , ε, T . Based on the local existence of solution, the uniform upper estimates (Lemma 3.2) and the lower bound estimate of density (3.24), we can conclude the following theorem.…”

“…Based on the uniform estimates of density, we can further investigate the global existence and large time behavior of entropy solutions. The idea can be applied to the other hyperbolic systems of conservation laws with source terms, see [15] for the multi-dimensional compressible Euler equations with spherical symmetry.…”

Existence and Large Time Behavior of Entropy Solutions to One-Dimensional Unipolar Hydrodynamic Model for Semiconductor Devices with Variable Coefficient Damping

“…From (20), the velocity u = m/ρ is uniformly bounded, i.e., |u | <C. Then, following the same way of [20], we could obtain…”

Global solutions to rotating motion of isentropic flows with cylindrical symmetry

“…Chen [3] proved a global existence theorem with large L ∞ data having only non-negative initial velocity. Recently, Huang-Li-Yuan [17] proved the global existence of weak solutions of the isentropic Euler equations with spherical symmetry including the origin by L ∞ compensated compactness framework and the vanishing viscosity method. Chen et al [9,10] proved the global existence of finite-energy entropy solution by using L p compensated compactness framework.…”