Abstract
We prove uniqueness in the class of integrable and bounded nonnegative solutions in the energy sense to the Keller-Segel (KS) chemotaxis system. Our proof works for the fully parabolic KS model, it includes the classical parabolic-elliptic KS equation as a particular case, and it can be generalized to nonlinear diffusions in the particle density equation as long as the diffusion satisfies the classical McCann displacement convexity condition. The strategy uses Quasi-Lipschitz estimates for the chemoattractant equation and the above-the-tangent characterizations of displacement convexity. As a consequence, the displacement convexity of the free energy functional associated to the KS system is obtained from its evolution for bounded integrable initial data.
Highlights
The classical Keller-Segel (KS) model for chemotaxis is the system∂tn = κ∆n − χ div (n∇c), ∂tc = η∆c + θn − γc.Here, n is the number/mass density of a bacteria/cell population and c represents the concentration of a chemical attractant that can suffer chemical degradation and that it is produced by the cells themselves due to chemotactic interaction
Concerning the fully parabolic KS system, we find global in time solutions satisfying all properties stated in Definition 1.2 except the L∞ bounds in [15] for d = 2 and the subcritical mass case m < 8π
In [16] we find that if the forcing term of the heat equation has bounded mean oscillation (BMO), still with respect to the parabolic metric, than the same holds true for second order space derivatives and first order time derivatives of the solution
Summary
The boundedness of the density implies that we have a uniform in bounded time intervals estimate on the quasi-Lipschitz constant of part of the velocity field These are the basic properties that imply the uniqueness for bounded solutions. L∞-apriori estimates were obtained in the classical parabolic-elliptic KS equation ε = 0 with d ≥ 2 and α ≥ 0 for small Ld/2 initial data in [20, 21] These results together with similar arguments as in [12] to get the free energy dissipation property and the Fisher information bounds, could lead to the existence of bounded solutions in these cases. Concerning the fully parabolic KS system, we find global in time solutions satisfying all properties stated in Definition 1.2 except the L∞ bounds in [15] for d = 2 and the subcritical mass case m < 8π.
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