WIAS Preprint No. 2967, (2022)

Hyperbolic-parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion


  • Druet, Pierre-Étienne
    ORCID: 0000-0001-5303-0500
  • Hopf, Katharina
    ORCID: 0000-0002-6527-2256
  • Jüngel, Ansgar
    ORCID: 0000-0003-0633-8929

2020 Mathematics Subject Classification

  • 35M11 35L45 35K40 35A09


  • Hyperbolic-parabolic systems, cross diffusion, quasilinear second-order symmetric systems, initial-value problem, entropy structure




We investigate degenerate cross-diffusion equations with a rank-deficient diffusion matrix that are considered to model populations which move as to avoid spatial crowding and have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic-parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in H^s(mathbbT^d) for s>d/2+1.

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