Global solutions to cross-diffusion systems with independent advections in one dimension

TL;DR

This paper introduces a new unified method to construct solutions for one-dimensional cross-diffusion systems with advection, covering all pressure exponents and initial data, including the first existence result for the case  > 1.

Contribution

A novel approach using vanishing viscosity and correlation of oscillations to establish solutions without structural assumptions for all pressure exponents.

Findings

Constructed solutions as limits of viscous approximations.

Identified limits of nonlinear terms via correlated oscillations.

Achieved the first existence result for  > 1 without structural assumptions.

Abstract

We consider cross-diffusion systems describing evolution of two species $u$ and $v$ moving according to Darcy's law with the pressure law $p(s) = \frac{1}{\alpha-1} s^{\alpha-1}$ where $s=u+v$. One of the most challenging questions in the field is the construction of solutions to the problem in the presence of additional advection fields, without imposing any artificial structure on the fields or the initial conditions. Although advection arises naturally in these models, it breaks the symmetry of the system and prevents application of techniques developed in recent years. Here, we provide a new approach to construct solutions in one space dimension that works in a unified manner for all pressure exponents $\alpha \in (0,\infty)$ and for arbitrary initial data. In~particular, in the regime $\alpha > 1$, this yields the first existence result of its kind, obtained without any structural assumptions. We construct the solutions as a limit of a vanishing viscosity approximation $(u_{\varepsilon}, v_{\varepsilon})$. The main challenge is to identify the limit of $u_{\varepsilon} \, \partial_x p(s_{\varepsilon})$, where $s_{\varepsilon} = u_{\varepsilon} + v_{\varepsilon}$. The key new insight is that possible oscillations of $u_\varepsilon$ and $\partial_x p(s_\varepsilon)$ are correlated, simplifying the Young measure analysis in the compensated compactness argument and allowing identification of the limit. Somewhat surprisingly, in contrast to the theory of $2\times2$ hyperbolic systems, the argument relies on only three entropy-entropy flux pairs. This is particularly useful for $\alpha>2$, where it is unclear whether additional entropies are available.