The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

Abstract

We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the ideas of characteristic decomposition and the bootstrap method, we show that the solution is uniformly C 1, 1 6 up to the degenerate sonic boundary and that the sonic curve is C 1, 1 6 .

Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

“…We present the derivations of the second and third equations in (22), the other two equations are analogous. Making use of the definition of H and the commutator relation between∂ 0 and∂ + in (21), one finds by the first two equations of (15) that which is the second equation of (22).…”

“…We next derive the equation for X. Applying the commutator relation betweeñ ∂ − and∂ + in (21) gives…”

“…There are more related references on classical methods for solutions [17,18], on perturbation arguments and linear theory [10,33,36] and on asymptotic models [2,13], etc. In the past few years, the semi-hyperbolic patch problems, a kind of degenerate hyperbolic problems with sonic curves as boundaries, have been explored by applying the idea of characteristic decomposition, see [20,21,30,37] for the Euler system, [38,42] for the pressure-gradient system and [22] for the nonlinear wave system. The characteristic decomposition is a powerful tool for studying some degenerate hyperbolic problems which was revealed by Dai and Zhang [15], see [12,24,27,28,35] for more applications.…”

Existence of solutions to gas expansion problem through a sharp corner for 2‐D Euler equations with general equation of state

“…In recent years, a lot of significant work has been done for the 2-D compressible Euler system as well as numerous other related models for a variety of initial and boundary value problems; see, namely, Refs. [5][6][7][8][9][10][11][12][13][14][15][16]. In particular, for gas expansion problems through a sharp corner or wedge, we refer the reader to Refs.…”

On the existence and regularity of solutions of semi-hyperbolic patches to 2-D Euler equations with van der Waals gas

“…The regularity of solution of the semi-hyperbolic patch problem for the pressure-gradient system was discussed in [15] and for isentropic Euler system in [16]. The regularity results for isothermal Euler equations have been discussed by Hu et al [17]. The regularity results obtained for isentropic Euler system in [16] were improved by Hu et al in [18].…”