For the two-dimensional isentropic Euler equations with the Chaplygin pressure law p(ρ)=−1/ρp(\rho)=-1/\rho, the Riemann initial data ρ−=ρ+=1\rho_- = \rho_+ = 1, v−=(−1,0)\mathbf{v}_- =(-1,0), v+=(1,0)\mathbf{v}_+=(1,0) — a jump only in the velocity component tangential to the interface — admit a classical self-similar solution consisting of a single stationary contact discontinuity. Březina, Kreml and Mácha (2021) proved non-uniqueness of bounded admissible weak solutions for Chaplygin Riemann data leading to δ\delta-shocks or to two contact discontinuities, but explicitly left open whether this stationary contact discontinuity is unique among bounded admissible weak solutions, noting that the analogous incompressible vortex-sheet solution is non-unique (Székelyhidi 2011). The question has since been answered — non-uniqueness — for every strictly increasing pressure law taking positive values (Krupa–Székelyhidi 2025 for a computer-designed law; Horimoto 2026 for all such laws), but those hypotheses exclude the negative Chaplygin pressure, whose energy density 1/(2ρ)1/(2\rho) is decreasing. The surviving open core is whether the stationary contact discontinuity for p(ρ)=−1/ρp(\rho)=-1/\rho is unique or wildly non-unique in the class of bounded admissible weak solutions.

Background

The isentropic Euler equations for a compressible inviscid gas in two space dimensions read ∂tρ+div⁡(ρv)=0,∂t(ρv)+div⁡(ρv⊗v)+∇p(ρ)=0,\partial_t \rho + \operatorname{div}(\rho \mathbf{v}) = 0, \qquad \partial_t(\rho \mathbf{v}) + \operatorname{div}\big(\rho \mathbf{v}\otimes\mathbf{v}\big) + \nabla p(\rho) = 0, where ρ>0\rho>0 is the density, v\mathbf{v} the velocity, and p(ρ)p(\rho) the pressure law. The Chaplygin gas is the pressure law p(ρ)=−1/ρp(\rho)=-1/\rho, studied since Chaplygin's 1904 work on gas jets and used both in high-speed aerodynamics and as a dark-energy model in cosmology; mathematically it is a singular negative, concave pressure for which both characteristic fields are linearly degenerate (each characteristic wave speed is constant along its own wave family, so no smooth nonlinear waves form) and classical solutions break down violently, the one-dimensional Riemann solver producing δ\delta-shocks — measure-valued solutions carrying a Dirac mass in the density — instead of classical shocks for data with converging normal velocity.

A weak solution solves the system in the sense of distributions; it is called admissible if it additionally satisfies the energy inequality, meaning the total energy density ρε(ρ)+ρ∣v∣2/2\rho\varepsilon(\rho)+\rho|\mathbf{v}|^2/2, with internal energy determined by ε′(ρ)=p(ρ)/ρ2\varepsilon'(\rho)=p(\rho)/\rho^2, obeys the inequality form of energy conservation. The Riemann problem prescribes initial data that are constant on each half-plane x2>0x_2>0 and x2<0x_2<0. When the jump between the two states occurs only in the velocity component tangential to the interface (density and normal velocity continuous), the classical one-dimensional self-similar solution is a single contact discontinuity — the compressible analogue of the vortex sheet datum of incompressible hydrodynamics, whose instability is the Kelvin–Helmholtz mechanism.

The convex-integration theory of De Lellis and Székelyhidi (2009, 2010) produces bounded admissible weak solutions of Euler systems that are highly non-unique (so-called wild solutions). Applied to the compressible Riemann problem, starting with Chiodaroli–De Lellis–Kreml (2015) for p(ρ)=ρ2p(\rho)=\rho^2, a line of work established for polytropic laws p(ρ)=ργp(\rho)=\rho^\gamma that non-uniqueness holds whenever the one-dimensional solution contains a shock, while data whose solution consists only of rarefaction waves are unique (Chen–Chen 2007; Feireisl–Kreml 2015). Contact-discontinuity data — in particular a single stationary contact discontinuity — lay outside all of these results, for every pressure law.

For the Chaplygin gas, Březina–Kreml–Mácha (2021) proved that two-dimensional Riemann data whose one-dimensional solution is a δ\delta-shock, or consists of two contact discontinuities (which requires a nonzero jump in the normal velocity, v−2−v+2>0v_-^2 - v_+^2 > 0), give rise to infinitely many bounded admissible weak solutions. Their method constructs admissible fan subsolutions — piecewise-constant self-similar subsolutions whose feasibility reduces to a finite algebraic system of Rankine–Hugoniot conditions (the jump conditions expressing conservation across each interface) and inequalities, feeding into the convex-integration machinery. They point out that for the stationary datum ρ−=ρ+=1\rho_- = \rho_+ = 1, v±=(±1,0)\mathbf{v}_\pm = (\pm 1, 0) (jump only in the tangential velocity, so v−2=v+2=0v_-^2 = v_+^2 = 0, outside their theorem's range) the one-dimensional solution is a single stationary contact discontinuity, and the question of its uniqueness among bounded admissible weak solutions was open — while in the incompressible case the corresponding vortex-sheet solution was already known to be non-unique (Székelyhidi 2011).

The contact-discontinuity question has very recently been resolved for positive pressure laws. Krupa–Székelyhidi (2025) developed a computer-assisted fan-subsolution framework that treats the pressure law itself as a degree of freedom and found a smooth pp with p>0p>0, p′>0p'>0 for which this exact datum yields infinitely many bounded admissible weak solutions; they also proved that one-dimensional bounded solutions for contact data are unique for any such law, so the non-uniqueness is genuinely two-dimensional. Horimoto (2026, preprint) then constructed admissible fan subsolutions for the same symmetric datum for every strictly increasing pressure law p∈C1((0,∞);(0,∞))p\in C^1((0,\infty);(0,\infty)) with positive values, covering all polytropic laws. Both results require p>0p>0. The Chaplygin law is strictly increasing but negative, and its internal-energy density (pressure potential) P(ρ)=ρε(ρ)=1/(2ρ)P(\rho)=\rho\varepsilon(\rho)=1/(2\rho) is decreasing in ρ\rho, in contrast to the increasing energy density of positive laws, so the known algebraic parameter regimes do not transfer verbatim; the Chaplygin instance of the question remains unaddressed in the literature.

Problem Statement

Consider the two-dimensional isentropic Euler system with the Chaplygin pressure law p(ρ)=−1/ρp(\rho) = -1/\rho and Riemann initial data ρ−=ρ+=1\rho_- = \rho_+ = 1 with v−=(−1,0)\mathbf{v}_- = (-1, 0) for x2<0x_2 < 0 and v+=(1,0)\mathbf{v}_+ = (1, 0) for x2>0x_2 > 0, i.e. a jump only in the velocity component tangential to the interface. The classical one-dimensional self-similar solution of this Riemann problem is a single stationary contact discontinuity along x2=0x_2 = 0. Determine whether this solution is unique in the class of bounded admissible weak solutions (weak solutions satisfying the energy inequality, with the internal energy fixed by ε′(ρ)=p(ρ)/ρ2\varepsilon'(\rho) = p(\rho)/\rho^2) for this pressure law: either construct infinitely many bounded admissible weak solutions emanating from this initial datum, or prove that the stationary contact discontinuity is the unique bounded admissible weak solution. This is the Chaplygin-gas instance of the compressible vortex-sheet uniqueness question; the corresponding question has been answered — non-uniqueness — for every strictly increasing pressure law taking positive values, but the negative Chaplygin pressure is not covered by those results. No restriction to self-similar, one-dimensional, or other subclasses of weak solutions is part of the question: a uniqueness claim must hold for all bounded admissible weak solutions, and a non-uniqueness claim may be established by any construction.

References

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