Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion

Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions $d\geq 2$ and in all of space for $d\geq 3$, the uniqueness being a result previously not known for PKS with degenerate diffusion. We generalize the notion of criticality for PKS and show that subcritical problems are globally well-posed. For a fairly general class of problems, we prove the existence of a critical mass which sharply divides the possibility of finite time blow up and global existence. Moreover, we compute the critical mass for fully general problems and show that solutions with smaller mass exists globally. For a class of supercritical problems we prove finite time blow up is possible for initial data of arbitrary mass.Comment: 31 page

Interpolation inequalities for weak solutions of nonlinear parabolic systems

Abstract The authors investigate differentiability of the solutions of nonlinear parabolic systems of order 2 m in divergence form of the following type ∑ | α | ≤ m ( - 1 ) | α | D α a α X , D u + ∂ u ∂ t = 0 . The achieved results are inspired by the paper of Marino and Maugeri 2008, and the methods there applied. This note can be viewed as a continuation of the study of regularity properties for solutions of systems started in Ragusa 2002, continued in Ragusa 2003 and Floridia and Ragusa 2012 and also as a generalization of the paper by Capanato and Cannarsa 1981, where regularity properties of the solutions of nonlinear elliptic systems with quadratic growth are reached. Mathematics Subject Classification (2000) Primary 35K41, 35K55. Secondary 35B65, 35B45, 35D1

A gradient flow approach to the Keller-Segel systems

These notes are dedicated to recent global existence and regularity results on the parabolic-elliptic Keller-Segel model in dimension 2, and its generalisation with nonlinear diffusion in higher dimensions, obtained throught a gradient flow approach in the Wassertein metric. These models have a critical mass Mc such that the solutions exist globally in time if the mass is less than Mc and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out

A gradient flow approach to the Keller-Segel systems

These notes are dedicated to recent global existence and regularity results on the parabolic-elliptic Keller-Segel model in dimension 2, and its generalisation with nonlinear diffusion in higher dimensions, obtained throught a gradient flow approach in the Wassertein metric. These models have a critical mass Mc such that the solutions exist globally in time if the mass is less than Mc and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out

Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction

In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard system with singular potential, degenerate mobility, and a reaction term. In particular, we prove the existence of a global attractor with finite fractal dimension, the existence of an exponential attractor, and convergence to equilibria for two physically relevant classes of reaction terms