Two-dimensional Riemann problem with three constant states for the system $u_t+(u^2)_x+(uv)_y=0$, $v_t+(uv)_x+(v^2)_y=0$: initial-data classes beyond three contact discontinuities

For the two-dimensional $2\times 2$ hyperbolic system $u_t+(u^2)_x+(uv)_y=0$, $v_t+(uv)_x+(v^2)_y=0$ of nonlinear conservation laws, Pang, Tian and Yang (2013) classified and explicitly constructed the global self-similar solutions for general three-constant Riemann initial data in the special class where all three initial discontinuities are contact discontinuities (two structural types, six exact configurations involving shocks, rarefactions, a spiral interaction domain, and a delta shock). They explicitly deferred the remaining classes of three-constant data, in which the initial jumps project other combinations of planar elementary waves (shocks, rarefactions, delta shocks, or two waves per jump). Later work has touched those deferred classes only partially: Shen (2015) constructed, under the one-planar-wave-per-jump assumption (H), the nine global solutions for the special right-angle three-piece geometry when exactly one of the three interfaces projects a planar delta shock, and explicitly deferred the remaining configurations. The completion of this classification for general three-constant data remains open.

Background

A multidimensional Riemann problem for a system of conservation laws asks for the solution when the initial data are piecewise constant, with the constant states occupying sectors of the plane separated by rays emanating from one point. While the one-dimensional Riemann problem, with two constant states separated by a single point, is classical, in two space dimensions there is no general existence or classification theory, and even the conjectured catalogue of possible wave-interaction patterns for the compressible Euler equations (the Zhang-Zheng conjecture of 1990) remains unproved; progress therefore proceeds case by case on simplified model systems. One of the earliest and most studied models is the nonstrictly hyperbolic $2\times 2$ system $u_t+(u^2)_x+(uv)_y=0,\qquad v_t+(uv)_x+(v^2)_y=0,$ derived as a simplified analogue of the two-dimensional Euler equations and first analyzed systematically by Tan and Zhang. Here $u$ and $v$ are the two conserved quantities; the system is hyperbolic in the sense that its characteristic wave speeds, the eigenvalues of the flux Jacobians, are real in every spatial direction, and nonstrictly hyperbolic because these speeds may coincide.

As in one dimension, the elementary waves are planar shocks (S), centered rarefaction waves (R), and contact discontinuities (J): a planar shock is a straight discontinuity front across which $(u,v)$ jumps, traveling at a speed fixed by the Rankine-Hugoniot conditions, the algebraic jump relations expressing conservation of each quantity across a moving front; a centered rarefaction wave is a continuous fan in which straight characteristics spread apart and the solution interpolates smoothly between two constant states; and a contact discontinuity is a discontinuity traveling at the characteristic speed of a linearly degenerate field, one whose wave speed is constant along its integral curves, so that the wave neither compresses into a shock nor spreads into a fan. Because the system is nonstrictly hyperbolic and has linearly degenerate structure, Riemann solutions may also contain delta shocks ($\delta$), singular measure-valued waves that carry a Dirac mass of the conserved quantity. Because the initial data and the system are invariant under the scaling $(x,y,t)\mapsto(\lambda x,\lambda y,\lambda t)$, solutions are sought as functions of the self-similar variables $(\xi,\eta)=(x/t,y/t)$; the reduced system is of mixed type, hyperbolic far from the origin and elliptic (its two characteristic speeds become complex) in a neighborhood of the origin. Most constructions in this literature, including all four-quadrant (four-constant) results of Tan and Zhang, work under assumption (H): away from a neighborhood of the origin, each initial jump projects exactly one planar elementary wave, meaning that the one-dimensional Riemann problem posed by its two adjacent states along the jump direction is solved by that single planar wave. Under (H) the four-constant problem was resolved class by class (four-J data and data involving rarefactions, with further combinations deferred), yielding dozens of exact configurations.

For three-constant initial data, Pang, Tian and Yang first treated the special geometry in which the three separation rays lie along the positive $x$-axis, positive $y$-axis, and negative $x$-axis. Their 2013 paper then treated general three-constant data, in which the constant states $(u_1,v_1)$, $(u_2,v_2)$, $(u_3,v_3)$ occupy three sectors with an arbitrary opening angle $\alpha$, but only in the class where all three initial discontinuities are contact discontinuities. In that class the admissible data split into two structural types, $J_{12}^-J_{23}^-J_{31}^-$ and $J_{12}^-J_{23}^+J_{31}^+$, where $J_{ij}^{\pm}$ denotes a contact discontinuity separating the constant states $i$ and $j$, the superscript distinguishing its two possible orientations; generalized characteristic analysis, which constructs the solution from the characteristic curves of the self-similar system (in particular by solving Goursat-type problems, boundary-value problems with data prescribed on two intersecting characteristic curves, at the wave-interaction points), yields six exact self-similar configurations, including a spiral-like solution on a domain bounded by a closed chain of characteristic curves constructed through a monotone fixed-point argument (a monotone iteration over nested candidate curves converging to the limiting characteristic), shocks penetrating entire rarefaction fans, and a configuration containing a delta shock. Numerical computations with a second-order nonoscillatory central finite-volume scheme confirmed five of the six configurations sharply; the numerical picture of the most complex case (Case 2.3) was acknowledged to be not very clear. A parallel line of work studies sibling $2\times 2$ systems (for example the Hwang-Lindquist system $u_t+(u^2)_x+(u^2)_y=0$, $\rho_t+(\rho u)_x+(\rho u)_y=0$ of isotropic-media elasticity), for which the analogous three-piece and four-quadrant classifications have since been completed without the one-wave-per-jump restriction.

Problem Statement

For the two-dimensional hyperbolic system of nonlinear conservation laws $u_t+(u^2)_x+(uv)_y=0,\qquad v_t+(uv)_x+(v^2)_y=0,$ consider the three-constant Riemann initial data $(u,v)_{t=0}=\begin{cases}(u_1,v_1), & x>0,\ y>0,\\ (u_2,v_2), & x<0,\ y>x\tan\alpha,\\ (u_3,v_3), & \text{otherwise},\end{cases}$ with $\alpha\in(\pi/2,\pi)\cup(\pi,3\pi/2)$, in which the three constant states occupy three sectors meeting at the origin. Pang, Tian and Yang (2013) resolved the case in which all three initial discontinuities are contact discontinuities. Complete the remaining classification: determine all admissible classes of such three-constant data in which the initial jumps are not exactly three contact discontinuities — that is, classes in which one or more initial jumps project a planar shock, a centered rarefaction wave, a delta shock, or a combination of two planar elementary waves (an initial jump projects a planar wave when the one-dimensional Riemann problem for its two adjacent states, taken along the direction normal to the jump, is solved by that planar wave or wave pair) — and, for every such class, construct the exact global self-similar solution in the $(\xi,\eta)=(x/t,y/t)$ plane. Here a datum is admissible if each of its three initial jumps projects, in the sense just defined, entropy-admissible planar elementary waves of the listed types — a single shock, rarefaction, contact discontinuity, or delta shock, or a pair of two such waves — with shock and delta-shock admissibility given by the entropy conditions of this literature ("three incoming, one outgoing" for shocks; "none outgoing" for delta shocks).

The task is to (i) enumerate the admissible data classes and give explicit algebraic criteria on the constant states $(u_i,v_i)$ and the angle $\alpha$ that decide which class a given datum belongs to, with the classes mutually exclusive and exhaustive over the admissible data; and (ii) for each class, determine the full self-similar wave configuration — the types, locations, and interaction loci of all resulting shocks, rarefaction waves, contact discontinuities, and delta shocks, including any secondary waves generated at interaction points and the terminal singular points at which penetrating waves vanish — constructed rigorously, with the Rankine-Hugoniot conditions (the algebraic jump relations expressing conservation across a discontinuity front), the entropy conditions (the admissibility criteria selecting the physically relevant weak solution), and, where delta shocks occur, the generalized delta-shock relations (the Rankine-Hugoniot-type relations governing a measure-valued front and the mass it carries) all satisfied.

References

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