Two-dimensional Riemann problem for the canonical linearly degenerate system of conservation laws: removing the single-contact restriction for global self-similar solutions

The two-dimensional Riemann problem — piecewise-constant initial data whose constant states occupy sectors separated by rays from the origin — is largely open even for 2x2 systems of hyperbolic conservation laws. Pang, Cai, and Zhao (J. Math. Phys. 57(6), 061501, 2016) constructed and classified global self-similar solutions for the canonical linear system $u_t+u_x+v_y=0$, $v_t-v_x+u_y=0$ with three constant states, under the structural assumption (H) that each initial ray projects exactly one planar contact discontinuity; the self-similar reduction is of mixed type, hyperbolic outside the sonic circle and elliptic inside, and the construction matches a Goursat-type interaction analysis in the hyperbolic region with an $L^2$-based elliptic boundary value problem on the unit disk. Remove assumption (H) and construct and completely classify the global self-similar solutions for general three-sector data, in which each ray generically projects two planar contact discontinuities.

Background

A Riemann problem for a system of hyperbolic conservation laws is an initial-value problem whose initial data are piecewise constant; in one space dimension its solution is a self-similar fan of elementary waves — shock waves, rarefaction waves, and contact discontinuities — and the one-dimensional theory is classical. In two space dimensions, with constant states occupying sectors separated by rays emanating from the origin, no comparable general theory exists, even for $2\times 2$ systems. Because the data are homogeneous of degree zero, solutions are sought in self-similar variables $(\xi,\eta)=(x/t,y/t)$, and the reduced system is frequently of mixed type: hyperbolic far from the origin, where planar elementary waves interact, and elliptic near it, where the interaction region must be described by an elliptic boundary value problem. This mixed structure is the same central obstacle as in the two-dimensional Riemann program for the compressible Euler equations and their subsystems (the pressure-gradient and potential-flow systems).

The source paper studies a canonical prototype: the linear, constant-coefficient, strictly hyperbolic, linearly degenerate system (strictly hyperbolic means two real, distinct characteristic speeds; linearly degenerate means the wave speeds do not depend on the solution, so shocks and rarefactions cannot form)

$$\begin{cases}u_{t}+u_{x}+v_{y}=0, \\ v_{t}-v_{x}+u_{y}=0 \end{cases}$$

called system (1.1) there. Since the system is linear, its only elementary waves are contact discontinuities (jump discontinuities transported along characteristics; the one-dimensional Riemann solution across a ray generically consists of two of them, one per characteristic family). After the self-similar transformation the reduced system is elliptic inside the sonic circle $\xi^2+\eta^2=1$ and hyperbolic outside. For three constant states separated by three rays, under the structural assumption (H) — each initial ray outside a neighborhood of the origin projects exactly one planar contact discontinuity — the paper solves the interaction of planar contact discontinuities in the hyperbolic region as a Goursat problem (a hyperbolic boundary value problem with data prescribed on two characteristic curves issuing from a common point, treated by the generalized characteristic analysis method), proves existence and uniqueness of a weak, $L^2$-based solution of the elliptic boundary value problem on the unit disk, and thereby classifies the global solutions into five cases with explicit algebraic criteria. Predecessor works of the same group treated related three-piece and three-contact configurations, including anisotropic variants, and the two-dimensional Riemann program for $2\times 2$ systems in isotropic and anisotropic media goes back to Hwang and Lindquist.

The construction, however, is bounded by assumption (H).

Problem Statement

Remove assumption (H) and thereby extend the global self-similar theory of the two-dimensional Riemann problem beyond the restriction of the Pang-Cai-Zhao construction.

For system (1.1) $u_t+u_x+v_y=0$, $v_t-v_x+u_y=0$: for initial data consisting of three constant states in three sectors separated by rays from the origin, allow each initial ray to project two planar contact discontinuities (the generic one-dimensional Riemann resolution across a ray for this system). Construct the global self-similar solutions — covering both the hyperbolic interaction region and the elliptic core, across the sonic circle $\xi^2+\eta^2=1$ where the type changes — and completely classify the solution patterns with explicit criteria covering all configurations, extending the source paper's five-case classification obtained under assumption (H).

References

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