The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited

We study a class of parabolic quasilinear systems, in which the diffusion matrix is not uniformly elliptic, but satisfies a Petrovskii condition of positivity of the real part of the eigenvalues. Local well-posedness is known since the work of Amann in the 90s, by a semi-group method. We first revisit these results in the context of Sobolev spaces modelled on $L^2$ and then explore the endpoint Besov case $B_{p,1}^{d/p}$. We also exemplify our method on the SKT system, showing the existence of local, non-negative, strong solutions.

💡 Research Summary

The paper revisits the Cauchy problem for a class of quasilinear parabolic systems whose diffusion matrix is not uniformly elliptic but satisfies the Petrovskii condition—that the real parts of all eigenvalues are uniformly positive. The authors first re‑establish local well‑posedness in Sobolev spaces based on L², namely H^s(T^d) with s>d/2, and then extend the analysis to the endpoint Besov spaces B_{p,1}^{d/p} for any finite p≥1.

The model under consideration is
∂t U – Σ{k=1}^d ∂_k ( A(U) ∂k U ) = F, U|{t=0}=U₀,
with U: ℝ₊×T^d→ℝ^N, a smooth matrix‑valued map A:ℝ^N→M_N(ℝ) and a source term F. The linearised problem replaces A(U) by a fixed matrix M and serves as the backbone for the whole theory. The Petrovskii condition is formalised as B∈P_δ:= {B∈M_N(ℝ) : Re λ≥δ for all λ∈Sp(B)} for some δ>0. This condition, weaker than uniform ellipticity, still guarantees coercivity of the principal part and enables energy estimates.

The authors introduce the energy space
E_T^s = C⁰(