Uniqueness of strong solutions and weak–strong stability in a system of cross-diffusion equations

Abstract

Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in Ehrlacher and Bakhta (ESAIM Math Model Numer Anal, 2017). Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of non-negative strong solutions. The proof of the existence relies on the use of an appropriate linearized problem and a fixed-point argument. In addition, a weak–strong stability result is obtained for this system in dimension one which also implies uniqueness of weak solutions

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Acknowledgements

The work of MB has been supported by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse. MB and JFP acknowledge support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, Münster. VE acknowledges support by the ANR via the ANR JCJC COMODO project. VE and JFP are grateful to the DAAD/ANR for their support via the project 57447206. Furthermore, the authors would like to thank Robert Haller-Dintelmann (TU Darmstadt) for useful discussions. We would also like to thank the anonymous referee for his very useful comments and suggestions.

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Appendix: Microscopic interpretation

Appendix: Microscopic interpretation

Following [4], we briefly describe a lattice-based modelling approach and a formal way to obtain a (7) in the limit. We start with a one-dimensional lattice on which particles of \(i=1,\ldots n\) species can jump to neighbouring sites. Let \({\mathcal {T}}_h\) denote an equidistant grid of mesh size h, where a cell is either empty or can be occupied by at most one particle. We denote the probability to find a particle of species i at location x and time t by

$$\begin{aligned} c_i(x,t)&=P(\text {particle of species }i\text { at position }x\text { at time }t), \end{aligned}$$

and assume that the motion of these particles is due to two different effects: Diffusion and exchange (switching) of particles of different species. To this end, we introduce the rates

$$\begin{aligned} \Pi ^{+}_{c_i}&=P(\text {jump of }c_i\text { from position }x\text { to }x+h\text { in }(t,t+\Delta t)) \end{aligned}$$
(52)
$$\begin{aligned}&=K_{i0}(1-\rho ) + \sum _{j=1,\,i\ne j}^n K_{ij}c_j, \end{aligned}$$
(53)
$$\begin{aligned} \Pi ^{-}_{c_i}&=P(\text {jump of }c_i\text { from position }x\text { to }x-h\text { in } (t,t+\Delta t)) \nonumber \\&=K_{i0}(1-\rho ) + \sum _{j=1,\,i\ne j}^n K_{ij}c_j. \end{aligned}$$
(54)

Here, \(K_{i0}\) is a diffusion coefficient which controls the tendency of a particle to jump to a neighbouring site. Since we restrict to at most one particle per site, this has to be modified by a factor of \((1-\rho )\), i.e. the particle can only jump if the target site is empty. On the other hand, in order to exchange places with a particle from a different species, the target site has to be occupied and thus, for the second term we have to multiply the rate \(K_{ij}\) with \(c_j\).

Now we consider the following cases: If \(K_{i0} \gg K_{ij}\), then the probability of switching is small compared to that of diffusion and the effect of size exclusion will be essential. If, on the other hand, \(K_{i0} \ll K_{ij}\), switching will dominate and size exclusion will not play a role anymore. Note that in this case, \(\rho \), which is the sum of all densities, remains constant.

Our subsequent analysis deals with the case when \(K_{i0} \approx K_{ij}\), which is the most interesting. In fact, let us rewrite (52) as follows:

$$\begin{aligned} \Pi ^{+}_{c_i}&=K_{i0}(1-\sum _{j=1,i\ne j}^n c_j - c_i) + \sum _{j=1,\,i \ne j}^n K_{ij}c_j,\\&= K_{i0}(1- c_i) + \sum _{j=1,\,i\ne j}^n (K_{ij}-K_{i0})c_j. \end{aligned}$$

Now if \(K_{i0} \approx K_{ij}\), the switching will effectively anneal the size exclusion effect. In other words, it does not make a difference whether a target site is occupied by a particle of species j or if it is empty since in both cases, the particle at the source site can reach this target: either by jumping to the empty cell or by switching positions. The resulting PDE can be written as

$$\begin{aligned} \partial _t c_i&= \nabla \cdot (K_{i0}((1-c_i)\nabla c_i + c_i\nabla c_i + \sum _{j=1,i\ne j}^n (K_{i0}-K_{ij})(c_j \nabla c_i - c_i \nabla _j))\\&=\nabla \cdot (K_{i0}c_i + \sum _{j=1,i\ne j}^n (K_{i0}-K_{ij})(c_j \nabla c_i - c_i \nabla c_j)),\quad i=1,\ldots , n. \end{aligned}$$

which reveals that we are dealing with a perturbation of the heat equations, as already entailed in (3) in Introduction.

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Berendsen, J., Burger, M., Ehrlacher, V. et al. Uniqueness of strong solutions and weak–strong stability in a system of cross-diffusion equations. J. Evol. Equ. 20, 459–483 (2020). https://doi.org/10.1007/s00028-019-00534-4

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