Lai, Xiang, and Zhou (2018/2022) proved that fan-shaped piecewise-smooth solutions of the generalized Riemann problem for the compressible isothermal Euler equations in two and three space dimensions — consisting of two shocks and one contact discontinuity perturbed by amplitude ε\varepsilon from planar Riemann data — cannot exist globally in time, with lifespan upper bounds T(ε)≤Cε−2T(\varepsilon)\le C\varepsilon^{-2} in 2D and T(ε)≤exp⁡(Cε−1)T(\varepsilon)\le\exp(C\varepsilon^{-1}) in 3D, the latter conditional on an unestablished 3D local existence theory. The open core they leave behind is to complete the long-time theory of this wave pattern by (i) establishing or refuting local-in-time existence of piecewise-smooth solutions of the 3D generalized Riemann problem containing a nonlinear vortex sheet, and (ii) determining whether global stability is restored in a weak (entropy) solution class for perturbed planar shock–contact data.

Background

The inviscid compressible Euler equations ρt+∇⋅(ρu)=0\rho_t+\nabla\cdot(\rho\mathbf{u})=0, (ρu)t+∇⋅(ρu⊗u)+∇p=0(\rho\mathbf{u})_t+\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u})+\nabla p=0 govern the density ρ\rho and velocity u\mathbf{u} of an ideal adiabatic gas; in the isothermal case the equation of state is p=ρp=\rho, so pressure and density coincide. The one-dimensional Riemann problem — data constant on either side of a point discontinuity — was solved in the classical theory of hyperbolic conservation laws, and in one space dimension small-BV global existence and L1L^1 stability theories are essentially complete. Its multidimensional analogue is a central open challenge. The generalized Riemann problem (GRP) studies Cauchy data that are smooth on either side of a smooth curve (in 2D) or surface (in 3D) and jump across it in the direction of a one-dimensional Riemann solution; the expected solution is piecewise smooth with a fan-shaped wave structure whose fronts are free boundaries. The fan considered here consists of two shock waves and one contact discontinuity: a vortex sheet, i.e. a free interface across which density is continuous but tangential velocity jumps, so the front is a shear layer subject to the Kelvin–Helmholtz mechanism (amplification of interface perturbations by shear). Across each front the Rankine–Hugoniot conditions (mass and momentum conservation across a moving discontinuity) must hold. Local-in-time well-posedness of such free-boundary patterns is known only in pieces: Majda's 1983 memoirs established existence and stability of multidimensional shock fronts; Coulombel and Secchi (2008), building on the linear theory, proved nonlinear existence and uniqueness of compressible vortex sheets in two space dimensions by Nash–Moser iteration (an implicit-function scheme tolerating loss of derivatives, applicable because the linearized operator loses a finite number of derivatives), but only under a supersonic weak-stability condition on the tangential velocity jump; Chen and Li (2014) handled the two-dimensional GRP with composite waves including a contact discontinuity. In three space dimensions no local existence theory for the GRP with a nonlinear vortex sheet exists. For smooth (front-free) data, Sideris (1985) showed that 3D compressible Euler solutions form singularities in finite time regardless of size, so the discontinuous/GRP setting is the natural one for long-time questions. The one available global result of this type is Godin's (1990): perturbed planar shock fronts for second-order hyperbolic conservation laws of potential-flow type (an irrotational, barotropic reduction of Euler to a scalar wave-type equation) exist globally in n≥5n\ge 5 space dimensions — evidence that perturbed planar shocks are not generically unstable in high dimensions. Lai, Xiang, and Zhou (2018, published 2022) proved the complementary negative result for the isothermal system in physical dimensions: under explicit sign conditions on exponentially weighted integrals of the initial perturbation of size ε\varepsilon, the fan-shaped piecewise-smooth GRP solution (i) cannot exist globally in 2D, with lifespan T(ε)≤Cε−2T(\varepsilon)\le C\varepsilon^{-2}, and (ii) in 3D, conditionally on any local existence, satisfies T(ε)≤exp⁡(Cε−1)T(\varepsilon)\le\exp(C\varepsilon^{-1}). These scalings match the known sharp lifespans of small-data solutions of genuinely nonlinear wave equations, and the authors state that global stability of these planar shocks must consequently be sought in a weak solution class (distributional solutions satisfying the entropy inequality, for which no non-classical framework exists in this multidimensional Riemann setting). The paper proves only upper bounds; the 3D local theory and the weak-class question are left open.

Problem Statement

For the compressible isothermal Euler equations (p=ρp=\rho) in two and three space dimensions, complete the long-time theory of the generalized Riemann problem whose initial data are smooth perturbations of size ε\varepsilon of planar Riemann data producing a fan of two shock waves and one contact discontinuity (vortex sheet). The scientific objective is to determine whether this canonical multi-dimensional wave pattern is well posed locally in 3D and globally in some solution class, resolving the conditional picture left by Lai–Xiang–Zhou: (1) 3D local theory: prove, or disprove, local-in-time existence and uniqueness of piecewise-smooth solutions of the three-dimensional generalized Riemann problem with the same fan-shaped wave structure (two shocks plus nonlinear contact discontinuity), for smooth perturbations of planar data, at least in the regime analogous to the two-dimensional supersonic weak-stability condition of Coulombel–Secchi under which the 2D theory of Chen–Li holds. (2) Global weak-class behavior: determine whether these perturbed planar shock–contact data generate global-in-time weak (entropy) solutions of the isothermal Euler system — i.e., whether the classical global instability proved by Lai–Xiang–Zhou is replaced by stability after passing to the weak class.

References

  1. Ning-An Lai, Wei Xiang, Yi Zhou, Global instability of multi-dimensional plane shocks for isothermal flow, Acta Mathematica Scientia 42 (2022) 887–902, DOI 10.1007/s10473-022-0305-7, https://doi.org/10.1007/s10473-022-0305-7 (preprint arXiv:1807.07386, https://arxiv.org/abs/1807.07386)
  2. Jean-François Coulombel, Paolo Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. 41 (2008) 85–139, DOI 10.24033/asens.2064, https://doi.org/10.24033/asens.2064
  3. Shuxing Chen, Dening Li, Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system, J. Differential Equations 257 (2014) 1939–1988, DOI 10.1016/j.jde.2014.05.027, https://doi.org/10.1016/j.jde.2014.05.027
  4. Andrew Majda, The stability of multidimensional shock fronts, Mem. Amer. Math. Soc. 41 (1983) no. 275, DOI 10.1090/memo/0275, https://doi.org/10.1090/memo/0275
  5. Andrew Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc. 43 (1983) no. 281, DOI 10.1090/memo/0281, https://doi.org/10.1090/memo/0281
  6. Paul Godin, Long time existence of a class of perturbations of planar shock fronts for second order hyperbolic conservation laws, Duke Math. J. 60 (1990) 425–463, DOI 10.1215/s0012-7094-90-06017-x, https://doi.org/10.1215/s0012-7094-90-06017-x
  7. Thomas C. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985) 475–485, DOI 10.1007/BF01210741, https://doi.org/10.1007/BF01210741
  8. Beixiang Fang, Feimin Huang, Wei Xiang, Feng Xiao, Persistence of the steady planar normal shock structure in 3-D unsteady potential flows, J. London Math. Soc. 107 (2023) 1692–1753, DOI 10.1112/jlms.12723, https://doi.org/10.1112/jlms.12723 (preprint arXiv:2108.09207, https://arxiv.org/abs/2108.09207)
  9. Beixiang Fang, Ya-Guang Wang, Qin Zhao, On multi-dimensional linear stability of planar shock waves for Chaplygin gases, Appl. Math. Lett. 102 (2020) 106085, DOI 10.1016/j.aml.2019.106085, https://doi.org/10.1016/j.aml.2019.106085
  10. Gui-Qiang G. Chen, Yaguang Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal. 187 (2007) 369–408, DOI 10.1007/s00205-007-0070-8, https://doi.org/10.1007/s00205-007-0070-8
  11. Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan, Stabilization effect of elasticity on three-dimensional compressible vortex sheets, J. Math. Pures Appl. 172 (2023) 105–138, DOI 10.1016/j.matpur.2023.01.005, https://doi.org/10.1016/j.matpur.2023.01.005
  12. Ning-An Lai, Nico Michele Schiavone, Lifespan estimates for the compressible Euler equations with damping via Orlicz spaces techniques, J. Evol. Equ. 23 (2023), article 65, DOI 10.1007/s00028-023-00918-7, https://doi.org/10.1007/s00028-023-00918-7
  13. Jinbo Geng, Ke Hu, Ning-An Lai, Manwai Yuen, Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions, Adv. Nonlinear Anal. 13 (2024), article 20240043, DOI 10.1515/anona-2024-0043, https://doi.org/10.1515/anona-2024-0043