Convergence of Lax–Friedrichs and Godunov schemes for $2\times2$ nonstrictly hyperbolic systems with an isolated umbilic point beyond the symmetric $(a,b)=(3,0)$ prototype

The compensated-compactness convergence theory for Lax–Friedrichs and Godunov approximate solutions of $2\times2$ nonstrictly hyperbolic conservation laws arising in oil recovery is currently proved, for fluxes with higher-order terms, only for one prototype: the quadratic normal form with Schaeffer–Shearer parameters $(a,b)=(3,0)$ perturbed by higher-order terms of order $l>7$ at the umbilic point, under the symmetry $f_v=g_u$ and the assumption that genuine nonlinearity fails only on the curve $f_v=0$. The problem asks to extend this convergence theorem to the general admissible class: other pairs $(a,b)$ with $a\ne 1+b^2$ from the Schaeffer–Shearer classification, higher-order terms of the lowest order the analysis permits (determining whether the threshold $l>7$ is sharp), and fluxes without the structural symmetry and single-curve degeneracy hypotheses. The question concerns the documented gaps between the convergence theorem proved in Djoufedie–Felaco–Rubino–Sampalmieri (Continuum Mech. Thermodyn. 28, 2016, DOI 10.1007/s00161-015-0432-7) and the general admissible class, which remain open.

Background

A system of conservation laws $\partial_t U+\partial_x F(U)=0$ for $U=(u,v)$ is strictly hyperbolic at a state when the Jacobian $DF(U)$ has two real, distinct eigenvalues. In models of three-phase flow in porous media — for example water-alternating-gas (WAG) oil recovery — the reduced $2\times2$ flux loses strict hyperbolicity at isolated states, called umbilic points, where the two characteristic speeds coincide; near such a point the system is resonant and the standard theory of strictly hyperbolic systems (Glimm schemes, Lax shock conditions in their usual form) does not apply directly. Schaeffer and Shearer (1987) classified the quadratic parts of $2\times2$ fluxes with an isolated umbilic point: after invertible changes of the unknowns, each equivalence class is represented by the gradient of the potential

$C(u,v)=\tfrac{1}{2}\big(\tfrac{a}{3}u^3+bu^2v+uv^2\big),$

with admissible parameters $(a,b)$, and the parabola $a=1+b^2$ separates regions of qualitatively different wave geometry (on the parabola itself the degeneracy locus of the normal form is a line rather than a single point, so the umbilic point is not isolated and the parabolic case lies outside the isolated-umbilic class considered here), while the physically derived oil-recovery quadratic parts lie on the side $a<1+b^2$ and the mathematically most-studied prototype $(a,b)=(3,0)$ lies on the opposite side $a>1+b^2$.

For the numerical side: the Lax–Friedrichs and Godunov schemes are the two classical conservative finite-difference methods for hyperbolic systems, and the convergence question is whether their approximate solutions, as the mesh is refined, converge to a weak solution of the limit system. For resonant, nonstrictly hyperbolic fluxes the main rigorous route is compensated compactness (Tartar, Murat, DiPerna): one shows that the family of approximate solutions is compact in $L^1_{\mathrm{loc}}$, or at least that every Young measure (the probability measure encoding the possible oscillations of the sequence at almost every point) reduces to a Dirac mass, using a rich family of entropy pairs — pairs $(\eta,q)$ with $\nabla q=\nabla\eta\,DF$ — whose weak entropy dissipation measures are controlled.

The program culminating in Djoufedie–Felaco–Rubino–Sampalmieri (2016) is: Kan (1991) analyzed the pure quadratic $(a,b)=(3,0)$ model; Rubino (1995) built a compactness framework and proved Lax–Friedrichs and Godunov convergence for a $2\times2$ nonstrictly hyperbolic system of this type; Chen and Kan (1995, 2001) developed the vanishing-viscosity and scheme-convergence theory for the canonical quadratic classes, with the entropy kernel governed by a singular Euler–Poisson–Darboux (EPD) equation — a linear hyperbolic equation whose coefficients degenerate along the characteristics — posed in Riemann invariants (coordinates constant along characteristic families). Felaco–Rubino–Sampalmieri (2013) added higher-order perturbations to the flux for the vanishing-viscosity problem, and Djoufedie–Felaco–Rubino–Sampalmieri (2016) did the same for the Lax–Friedrichs and Godunov schemes. There the system is

$\partial_t u+\partial_x f(u,v)=0,\qquad \partial_t v+\partial_x g(u,v)=0,$

with $f=\tfrac{3}{2}u^2+\tfrac{1}{2}v^2+\phi$, $g=uv+\psi$ and $\phi,\psi=O((|u|+|v|)^l)$ with $l>7$ near the origin, which is exactly the $(a,b)=(3,0)$ gradient normal form plus a flat perturbation. The proof uses the symmetry $f_v=g_u$, an isolated umbilic point at the origin, genuine nonlinearity (both characteristic fields nonlinear) failing only on the curve $f_v=0$, quasi-convex Riemann invariants (Riemann invariants whose level curves are convex), invariant regions $\Sigma_c$ (bounded regions of state space, depending on a parameter $c$, that the approximate solutions never leave), and an entropy analysis in which the perturbed entropy kernel solves a singular EPD equation whose Riemann function is compared with that of the unperturbed quadratic system in weighted Hölder norms; East-type entropies are obtained through a Goursat problem, a boundary-value problem with data prescribed on two intersecting characteristic curves.

Problem Statement

Extend the convergence theorem of Djoufedie–Felaco–Rubino–Sampalmieri (2016) — convergence of Lax–Friedrichs and Godunov approximate solutions, via the compensated-compactness framework and the entropy construction in Riemann invariants, for $2\times2$ nonstrictly hyperbolic systems with an isolated umbilic point arising in oil recovery — beyond the symmetric prototype, along the three documented gaps between the proved theorem and the general class:

(i) Other normal forms: extend the compactness framework and the entropy construction to admissible pairs $(a,b)\ne(3,0)$ with $a\ne 1+b^2$ from the Schaeffer–Shearer classification, in the presence of higher-order terms $\phi,\psi$ vanishing at the umbilic point; note that the physically derived oil-recovery quadratic parts lie on the side $a<1+b^2$, opposite to the studied $(3,0)$ prototype.

(ii) Order of the perturbation: determine whether the hypothesis $\phi,\psi=O((|u|+|v|)^l)$ with $l>7$ near the umbilic point is sharp for the convergence theorem itself, or prove the theorem under a weaker order condition (parts of the existing analysis require only lower order: Felaco–Rubino–Sampalmieri (2013) states that $l>7$ is the best obtainable with that particular technique and that its basic lemmas hold for $l>3$). Sharpness here concerns the convergence theorem itself and not the EPD/entropy-kernel technique: a lower-order counterexample is an admissible flux with perturbations of that order whose Lax–Friedrichs/Godunov approximate solutions fail to converge, for instance through non-Dirac Young measures, whereas a demonstration that the EPD/entropy-kernel technique cannot reach a lower order is a technique-level limitation separate from the sharpness of the theorem.

(iii) Structural hypotheses: determine whether analogous Lax–Friedrichs/Godunov convergence results hold without the symmetry condition $f_v=g_u$ and without the assumption that genuine nonlinearity fails only on the curve $f_v=0$, both of which are used in that proof to simplify the rarefaction-wave geometry and the singular structure of the entropy equation.

The common objective is a convergence theory for the general class of $2\times2$ nonstrictly hyperbolic systems with isolated umbilic point under the weakest hypotheses under which the compensated-compactness program applies; the three directions are the explicit gaps between the current theorem and that objective. The three directions are independent open sub-questions: a complete resolution of the stated objective requires closing all three gaps, while a result on any one direction is a meaningful partial contribution.

References

1. George Noel Djoufedie, Elisabetta Felaco, Bruno Rubino, Rosella Sampalmieri, Convergence of Lax–Friedrichs and Godunov schemes for a nonstrictly hyperbolic system of conservation laws arising in oil recovery, Continuum Mechanics and Thermodynamics 28 (2016) 331–349 (online 2015-05-08), DOI 10.1007/s00161-015-0432-7, https://doi.org/10.1007/s00161-015-0432-7
2. Elisabetta Felaco, Bruno Rubino, Rosella Sampalmieri, Global existence to the Cauchy problem for hyperbolic conservation laws with an isolated umbilic point, Quarterly of Applied Mathematics 71 (2013) 629–659, DOI 10.1090/s0033-569x-2013-01328-6, https://doi.org/10.1090/s0033-569x-2013-01328-6
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