Sharp threshold and complete solution structure of the bi-symmetric four-rarefaction-wave Riemann problem for the two-dimensional compressible Euler equations

The interaction of four rarefaction waves is a fundamental configuration of the two-dimensional Riemann problem for the compressible Euler equations. Li and Zheng (Commun. Math. Phys. 296 (2010) 303-321) constructed global continuous self-similar solutions for four orthogonal planar rarefaction waves in the bi-symmetric class when the initial rarefaction waves are large, i.e. when the sound-speed parameter $c_2$ of the initial states lies below a threshold $c_2^0(\gamma)$ produced by their phase-plane analysis. They left open the sharp threshold $c_2^*(\gamma)$ separating global continuous solutions from singularity formation, the quantitative location of the vacuum boundary $\xi=B(\eta)$ , the monotonicity of the characteristic inclination $\beta$ along the boundary curve $bd$ of the simple-wave patch, and the analytic mechanism of the transition to sonic/shock formation observed numerically. This problem asks for the complete quantitative theory: the sharp critical amplitude, the geometric structure of the solution (vacuum boundary, characteristic-boundary monotonicity) up to that threshold, and the failure mechanism beyond it. All of these issues remain open for the Euler system; only pressure-gradient-system analogues and numerical simulations exist.

Background

The two-dimensional Riemann problem for the compressible Euler equations asks for the solution whose initial data are constant in each quadrant of the plane, so that elementary planar waves (shocks, rarefaction waves, contact discontinuities) issue from the origin and interact; it is a decades-old core open problem of hyperbolic conservation laws. Because the data are scale-invariant, one seeks self-similar solutions depending only on $\xi=x/t$ and $\eta=y/t$ ; these satisfy a quasilinear system of mixed type, hyperbolic where the self-similar flow is (pseudo-)supersonic and elliptic inside the sonic circle, the locus where the flow speed equals the local sound speed.

A rarefaction wave is a smooth self-similar wave across which the gas expands and density and pressure decrease continuously. The present problem concerns the four-rarefaction configuration, in which the initial states are chosen so that each quadrant-interface discontinuity resolves into a planar rarefaction wave. In the bi-symmetric class the four constant states are symmetric with respect to both coordinate axes, producing four orthogonal planar rarefaction waves; for a polytropic gas with equation of state $p(\rho)=\rho^\gamma$ (density $\rho$ , adiabatic exponent $\gamma>1$ ), the data are parametrized by the sound speeds $c_1\ge c_2$ of the alternating initial states, so smaller $c_2$ means stronger (larger) initial rarefaction waves.

For a single pair of adjacent rarefaction waves, Li and Zheng (Arch. Ration. Mech. Anal. 193 (2009) 623-657) resolved the binary interaction using the hodograph transformation (exchanging dependent and independent variables) and characteristic decomposition (rewriting derivatives along the two characteristic families as ODE-type relations); a direct version of the method avoiding the hodograph was developed in the companion works of Li-Yang-Zheng (J. Differential Equations 250 (2011) 782-798) and Ji-Zheng (J. Math. Anal. Appl. 406 (2013) 4-14). For the full four-wave bi-symmetric problem, Li and Zheng (Commun. Math. Phys. 296 (2010) 303-321) constructed the global continuous, bounded, self-similar solution when the initial rarefaction waves are large, i.e. $c_2$ below a threshold $c_2^0(\gamma)$ arising from their phase-space analysis, resolving the binary interactions inside the interaction region and producing vacuum (zero density) at the center. The construction has explicit limits: the threshold is not sharp (the supremal value $c_2^*(\gamma)$ up to which global continuous solutions exist is unknown, and the thresholds are expected to be numerical functions of $\gamma$ ); the spatial location of the vacuum interface $\xi=B(\eta)$ created by the binary expansions has no quantitative estimate (not even the value $B(0)$ ); it is unknown whether the characteristic inclination $\beta$ is monotone along the plus-characteristic boundary curve $bd$ of the simple-wave patch (a patch in which the solution is constant along one family of characteristics); and numerical simulations such as those of Han-Li-Tang (Commun. Comput. Phys. 10 (2011) 577-609), using an adaptive GRP (generalized Riemann problem, a high-resolution shock-capturing) scheme over the full set of four-wave two-dimensional Riemann configurations, show transonic shock structures forming for other amplitudes and configurations, while the mechanism and parameter boundary separating global smooth solutions from shock and sonic-boundary formation remain analytically unresolved. For the simplified pressure-gradient system (the Euler system retaining only the pressure-gradient terms in the velocity equations), the bi-symmetric four-rarefaction interaction was constructed by Bang (2009, three and four waves) and by Li-Xiao (2012), but these analogues do not determine the Euler thresholds. A survey by G.-Q. Chen (Communications on Applied Mathematics and Computation 5 (2023) 1015-1052, first posted in 2022 as arXiv:2204.00692 under the earlier title Multidimensional Riemann Problems) states that the Li-Zheng problems are solved within their class while 'the other types of 2-D Riemann problems are still wide open'.

Problem Statement

Determine the complete quantitative theory of the bi-symmetric four-rarefaction-wave Riemann problem for the two-dimensional isentropic compressible Euler equations with polytropic equation of state $p(\rho)=\rho^\gamma$ , arising from the open issues left by the construction of Li and Zheng (2010):

(1) Sharp threshold. For the adiabatic exponents $\gamma$ covered by the source construction, determine the critical value $c_2^*(\gamma)$ , the supremum of the initial-state sound-speed parameter $c_2$ for which a global continuous (classical away from the planar wave fronts and the vacuum boundary), bounded, self-similar solution of the bi-symmetric four-rarefaction problem exists. Prove existence of the global solution for every $c_2<c_2^*(\gamma)$ , and, for $c_2$ beyond the threshold, prove the failure mechanism: establish by proof which of gradient blow-up, shock formation, or sonic-contact obstruction between adjacent simple-wave patches (or a provably equivalent failure) occurs, or give a provable two-sided characterization of that mechanism. Determining $c_2^*(\gamma)$ subsumes the relationship to the construction threshold $c_2^0(\gamma)$ of Li-Zheng; that relationship follows from the determination rather than standing as a separate deliverable.

(2) Structure and geometry of the solution in the interaction region. Provide a quantitative determination or rigorous estimate of the vacuum boundary $\xi=B(\eta)$ of the binary expansions (including the value $B(0)$ ), and prove or refute the monotonicity of the characteristic inclination $\beta$ along the plus-characteristic curve $bd$ of the simple-wave patch.

Parts (1) and (2) are jointly posed.

The further extension of the global construction to non-bi-symmetric initial data (for example Configuration A) and to regimes where shocks or sonic boundaries form is a further open direction beyond that construction and belongs to the general two-dimensional Riemann problem, which remains wide open; no completion boundary is imposed on that extension here.

References

Jiequan Li and Yuxi Zheng, Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations, Communications in Mathematical Physics 296 (2010), no. 2, 303-321. DOI: 10.1007/s00220-010-1019-6, https://doi.org/10.1007/s00220-010-1019-6
Jiequan Li and Yuxi Zheng, Interaction of Rarefaction Waves of the Two-Dimensional Self-Similar Euler Equations, Archive for Rational Mechanics and Analysis 193 (2009) 623-657. DOI: 10.1007/s00205-008-0140-6, https://doi.org/10.1007/s00205-008-0140-6
Jiequan Li, Zhicheng Yang, and Yuxi Zheng, Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations, Journal of Differential Equations 250 (2011) 782-798. DOI: 10.1016/j.jde.2010.07.009, https://doi.org/10.1016/j.jde.2010.07.009
Xiaomei Ji and Yuxi Zheng, Characteristic decouplings and interactions of rarefaction waves of 2D Euler equations, Journal of Mathematical Analysis and Applications 406 (2013) 4-14. DOI: 10.1016/j.jmaa.2012.05.035, https://doi.org/10.1016/j.jmaa.2012.05.035
Seunghoon Bang, Interaction of three and four rarefaction waves of the pressure-gradient system, Journal of Differential Equations 246 (2009) 453-481. DOI: 10.1016/j.jde.2008.10.001, https://doi.org/10.1016/j.jde.2008.10.001
Fengbai Li and Wei Xiao, Interaction of four rarefaction waves in the bi-symmetric class of the pressure-gradient system, Journal of Differential Equations 252 (2012) 3920-3952. DOI: 10.1016/j.jde.2011.11.010, https://doi.org/10.1016/j.jde.2011.11.010
Ee Han, Jiequan Li, and Huazhong Tang, Accuracy of the Adaptive GRP Scheme and the Simulation of 2-D Riemann Problems for Compressible Euler Equations, Communications in Computational Physics 10 (2011) 577-609. DOI: 10.4208/cicp.280410.300710a, https://doi.org/10.4208/cicp.280410.300710a
Mingjie Li and Yuxi Zheng, Semi-hyperbolic Patches of Solutions to the Two-dimensional Euler Equations, Archive for Rational Mechanics and Analysis 201 (2011) 1069-1096. DOI: 10.1007/s00205-011-0410-6, https://doi.org/10.1007/s00205-011-0410-6
Gui-Qiang G. Chen, Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems, Communications on Applied Mathematics and Computation 5 (2023), no. 3, 1015-1052. DOI: 10.1007/s42967-022-00210-4, https://doi.org/10.1007/s42967-022-00210-4 (preprint arXiv:2204.00692, posted 2022 under the earlier title Multidimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems, https://arxiv.org/abs/2204.00692)
Geng Lai, Global solutions to a class of two-dimensional Riemann problems for the Euler equations with a general equation of state, Indiana University Mathematics Journal 68 (2019), no. 5, 1409-1464. DOI: 10.1512/iumj.2019.68.7782, https://doi.org/10.1512/iumj.2019.68.7782
Gui-Qiang G. Chen, Mikhail Feldman, and Wei Xiang, Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann Problems with Shocks for the Isentropic Euler system, arXiv:2504.07034 (2025), https://arxiv.org/abs/2504.07034
Gui-Qiang G. Chen, Qin Wang, and Shengguo Zhu, Global solutions of a two-dimensional Riemann problem for the pressure gradient system, Communications in Pure and Applied Analysis 20 (2021), no. 7-8, 2475-2503. DOI: 10.3934/cpaa.2021014, https://doi.org/10.3934/cpaa.2021014 (preprint arXiv:2008.10635, https://arxiv.org/abs/2008.10635)