Existence of solutions to reaction cross diffusion systems. (English) Zbl 1527.35187
Summary: Reaction cross diffusion systems are a two species generalization of the porous media equation. These systems play an important role in the mechanical modeling of living tissues and tumor growth. Due to their mixed parabolic-hyperbolic structure, even proving the existence of solutions to these equations is challenging. In this paper, we exploit the parabolic structure of the system to prove the strong compactness of the pressure gradient in \(L^2\). The key ingredient is the energy dissipation relation, which, along with some compensated compactness arguments, allows us to upgrade weak convergence to strong convergence. As a consequence of the pressure compactness, we are able to prove the existence of solutions in a general setting and pass to the Hele-Shaw/incompressible limit in any dimension.
MSC:
| 35M11 | Initial value problems for PDEs of mixed type |
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 49N15 | Duality theory (optimization) |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
reaction-diffusion equation; cell growth model; hyperbolic-parabolic system; convex duality; energy dissipation relationReferences:
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