For a first-order quasilinear hyperbolic system νt+A(ν)νx=0\nu_t+A(\nu)\nu_x=0 with CkC^k coefficients, the regularity of the eigenvalues and eigenvectors of A(ν)A(\nu) as functions of the state ν\nu, and the existence of normalized coordinates in which the characteristic fields align with coordinate axes, are foundational structural facts underlying wave-interaction formulas and Cauchy-problem existence and blowup theory. These facts are classical for strictly hyperbolic systems (simple eigenvalues) and were extended by Dai (2007) to systems whose characteristic multiplicities are constant over the state domain. Dai's paper also exhibits umbilic-type degeneracies at which, when the multiplicity of the eigenvalues varies across the state space, the eigenvalues need not be differentiable and differentiable eigenvector frames and normalized coordinates need not exist. The open problem is to characterize structurally — by necessary and sufficient geometric, algebraic, or analytic conditions on A(ν)A(\nu) or on the flux Jacobians (∇f,∇g)(\nabla f,\nabla g) of a conservation law with A(ν)=(∇f(ν))−1∇g(ν)A(\nu)=(\nabla f(\nu))^{-1}\nabla g(\nu) — exactly which non-constant-multiplicity systems still admit eigenvalue labelings and eigenvector fields of the same regularity as the coefficients together with normalized coordinates, and to determine how umbilic-type degeneracies obstruct or permit such normalization.

Background

A first-order quasilinear hyperbolic system in one space dimension is

νt+A(ν)νx=0,\nu_t+A(\nu)\nu_x=0,

where ν\nu takes values in a domain Ω⊂Rn\Omega\subset\mathbb{R}^n and A(ν)A(\nu) is an n×nn\times n matrix; the system is called hyperbolic when A(ν)A(\nu) has nn real eigenvalues (the characteristic speeds) and a complete set of eigenvectors. It is strictly hyperbolic when all eigenvalues are simple. A system of conservation laws (f(ν))t+(g(ν))x=0(f(\nu))_t+(g(\nu))_x=0 can be rewritten in this form with A(ν)=(∇f(ν))−1∇g(ν)A(\nu)=(\nabla f(\nu))^{-1}\nabla g(\nu) wherever ∇f\nabla f is invertible. Two structural features of A(ν)A(\nu) control much of the theory. First, the regularity of the spectral data — the eigenvalue functions λi(ν)\lambda_i(\nu) and left/right eigenvector fields li(ν),ri(ν)l_i(\nu),r_i(\nu) — as functions of the state ν\nu: for strictly hyperbolic systems with A∈CkA\in C^k these are classically CkC^k, and this regularity underpins characteristic decompositions, John-type wave formulas for the formation of singularities, and the global existence and blowup theory of the Li–Kong school. Second, the existence of normalized coordinates: a CkC^k change of dependent variables ν=ν(ν~)\nu=\nu(\tilde\nu) in which the eigenvectors of each characteristic family depend only on the corresponding block of new variables, so that the characteristic distribution is straightened; many existence and lifespan theorems are formulated in, or secretly rely on, such coordinates. When eigenvalues have multiplicity p>1p>1 these facts fail to be automatic. For matrix families depending on several real parameters, the roots of the characteristic polynomial can lose differentiability precisely where roots coalesce: the eigenvalues of the smooth symmetric family with matrix (abb−a)\begin{pmatrix}a&b\\ b&-a\end{pmatrix} are ±a2+b2\pm\sqrt{a^2+b^2}, which are Lipschitz but not C1C^1 at points where a=b=0a=b=0. In the conservation-law literature such states, where two or more characteristic speeds coincide, are called umbilic points (by analogy with surface points where principal curvatures coincide); they occur generically in models such as three-phase flow in porous media and isotropic elasticity. Dai (J. Math. Anal. Appl. 327 (2007), 188–202) gave the systematic constant-multiplicity theory: if the eigenvalues satisfy λ1=⋯=λp<λp+1<⋯<λn\lambda_1=\cdots=\lambda_p<\lambda_{p+1}<\cdots<\lambda_n with the integer pp fixed over Ω\Omega and A∈CkA\in C^k, then the eigenvalues are CkC^k, globally defined CkC^k left and right eigenvectors exist satisfying the biorthonormal normalization lirj=δijl_i r_j=\delta_{ij} with riTrj=δijr_i^{\mathrm{T}}r_j=\delta_{ij} inside the multiple block, conservation laws with A=(∇f)−1∇gA=(\nabla f)^{-1}\nabla g admit CkC^k normalized coordinates (the proof verifies the Frobenius-type closure condition [ri,rj]∈span{r1,…,rp}[r_i,r_j]\in\mathrm{span}\{r_1,\dots,r_p\} inside the multiple block, where [ , ][\ ,\ ] is the Lie bracket of vector fields), and a constant-multiplicity field of a conservation law is necessarily linearly degenerate in the sense of Lax (∇λ⋅ri≡0\nabla\lambda\cdot r_i\equiv 0, i.e. the characteristic speed is constant along the integral curves of its own eigenvector field, so the family produces no shock-forming steepening). The same paper shows by an explicit example (its Example 1.1, an umbilic degeneracy) that when the multiplicity is allowed to vary over the state space, differentiability of eigenvalues, the choice of differentiable eigenvectors, and normalized coordinates can all fail. The multiparameter root-regularity literature delimits what can be true without structural hypotheses. Rellich proved that analytic one-parameter self-adjoint families admit analytic eigenvalue branches. For smooth curves of hyperbolic polynomials (one parameter), Alekseevsky–Kriegl–Losik–Michor and follow-ups showed the roots admit C2C^2 parameterizations but not better in general; Colombini–Orrù–Pernazza proved that if the coefficients are CrC^r in one parameter, where rr is the maximal root multiplicity, then the roots are C1C^1 (and C2C^2 under C2rC^{2r} coefficients); Rainer proved that definable (tame, in the sense of o-minimal geometry) C∞C^\infty curves of hyperbolic polynomials admit definable C∞C^\infty root parameterizations, with sharp order-of-contact conditions. In contrast, for multiparameter families — the situation of a state-dependent matrix A(ν)A(\nu) — Rainer showed that even real-analytic (more generally quasianalytic, i.e. drawn from Denjoy–Carleman smooth classes whose members are still determined by their Taylor jets at a point) hyperbolic polynomial families in general admit only SBVlocSBV_{\mathrm{loc}} roots (special functions of locally bounded variation, i.e. locally integrable functions whose distributional derivative is a measure with no Cantor part) whose gradients exist a.e. and lie in Lloc1L^1_{\mathrm{loc}}, and whose gradients cannot belong to any LpL^p, p>1p>1; quasianalytic parameterization of the roots is possible only after local blow-ups and power substitutions of the parameter space. Parusinski–Rainer determined the optimal Sobolev regularity of roots. These results show that coefficient-level regularity of spectral data at varying multiplicity fails generically in several parameters, but they neither characterize the exceptional families where it holds nor address eigenvector frames or PDE normalized coordinates. Meanwhile the umbilic-point literature for non-strictly hyperbolic conservation laws (e.g. the Schaeffer–Shearer classification for 2×22\times 2 systems arising in oil recovery) studies Riemann-problem dynamics near umbilics while presupposing, not characterizing, the available spectral regularity. This is the gap the problem addresses.

Problem Statement

Let Ω⊂Rn\Omega\subset\mathbb{R}^n be a domain and let A:Ω→Rn×nA:\Omega\to\mathbb{R}^{n\times n} be a CkC^k matrix-valued function (k≥1k\ge 1) that is hyperbolic: A(ν)A(\nu) has nn real eigenvalues and a complete set of eigenvectors for every ν∈Ω\nu\in\Omega. Say the characteristic multiplicity is non-constant on Ω\Omega if the pattern of eigenvalue coincidences of A(ν)A(\nu) changes across Ω\Omega — for instance λ1(ν)=λ2(ν)\lambda_1(\nu)=\lambda_2(\nu) on a proper (e.g. codimension-two, umbilic-type) subset of Ω\Omega while the two eigenvalues are distinct elsewhere. Determine necessary and sufficient structural conditions — geometric, algebraic, or analytic, imposed on A(ν)A(\nu) itself or on the flux Jacobians (∇f,∇g)(\nabla f,\nabla g) in the conservation-law case A(ν)=(∇f(ν))−1∇g(ν)A(\nu)=(\nabla f(\nu))^{-1}\nabla g(\nu), f,g∈Ck+1f,g\in C^{k+1} — under which the following three conclusions, known for constant multiplicity (Dai 2007) and for simple spectra, continue to hold in the presence of non-constant multiplicity: (i) the eigenvalues of A(ν)A(\nu) admit a labeling by CkC^k functions on all of Ω\Omega; (ii) there exist left and right eigenvector fields li,ri∈Ck(Ω)l_i,r_i\in C^k(\Omega) satisfying the normalization li(ν)rj(ν)=δijl_i(\nu)r_j(\nu)=\delta_{ij} for all i,ji,j, with riTrj=δijr_i^{\mathrm{T}}r_j=\delta_{ij} within each block of coinciding eigenvalues; (iii) for systems of conservation laws, there exists a CkC^k change of coordinates ν=ν(ν~)\nu=\nu(\tilde\nu) to normalized coordinates in the sense of the Kong–Li–Dai definition — an invertible transformation in which, in the ν~\tilde\nu-space, each characteristic trajectory through the origin coincides with the corresponding coordinate axis in the simple case, and, in the multiple-block case, the eigenvectors of each coinciding characteristic block depend only on the corresponding block of new variables — as guaranteed by Dai's theorem under constant multiplicity. For each of (i), (ii), (iii), determine necessary and sufficient conditions on the coefficient data A(ν)A(\nu) or (∇f,∇g)(\nabla f,\nabla g), not restatements of the conclusions themselves, and determine how umbilic-type degeneracies — points or sets where the number of coinciding eigenvalues changes — obstruct or permit these conclusions.

References

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