The boundedness-by-entropy method for cross-diffusion systems

Abstract

A novel principle is presented which allows for the proof of bounded weak solutions to a class of physically relevant, strongly coupled parabolic systems exhibiting a formal gradient-flow structure. The main feature of these systems is that the diffusion matrix may be generally neither symmetric nor positive semi-definite. The key idea of the principle is to employ a transformation of variables, determined by the entropy density, which is defined by the gradient-flow formulation. The transformation yields at the same time a positive semidefinite diffusion matrix, suitable gradient estimates as well as lower and/or upper bounds of the solutions. These bounds are a consequence of the transformation of variables and are obtained without the use of a maximum principle. Several classes of cross-diffusion systems are identified which can be solved by this technique. The systems are formally derived from continuous-time random walks on a lattice modeling, for instance, the motion of ions, cells, or fluid particles. 17 5. Proof of Theorem 4 24 6. Further results and open problems 28 Appendix A. Relations to non-equilibrium thermodynamics 32 Appendix B. Formal derivation of two-species population models 33 Appendix C. A variant of the Aubin compactness lemma 34 References 36

A cross-diffusion system derived from a Fokker–Planck equation with partial averaging

“…The compactness is obtained from the entropy estimate (7). This technique is similar to those employed in our works [8,13]. The novelty here is the (nontrivial) observation that the cross-diffusion system (1) possesses a convex Lyapunov functional, defined by (6).…”

“…However, we claim that this notion is appropriate since it naturally generalizes physical situations. For details, we refer to [8].…”

“…The novelty here is the (nontrivial) observation that the cross-diffusion system (1) possesses a convex Lyapunov functional, defined by (6). Moreover, compared to [8,13], we are facing additional technical difficulties due to the quotient u 1 /u 2 .…”

“…It holds clearly S(w, 0) = 0. Standard arguments show that S is continuous (see, e.g., the proof of Lemma 5 in [8]). Because of the compact embedding…”

“…These systems are analyzed in, e.g., [5,8], essentially for m = 2. In this application, p i is often given by the sum p i1 (u 1 ) + p i2 (u 2 ), and consequently, the results of [5,8] do not apply and we need to develop new ideas. Our first main result is the global-in-time existence of weak solutions to (1).…”