FINITE-TIME SINGULARITIES OF AN AGGREGATION EQUATION IN R WITH FRACTIONAL DISSIPATION

We consider an aggregation equation in Rn, n ≥ 2, with fractional dissipation, namely, ut + ∇ · (u∇K ∗ u) = −ν(−∆)γ/2u , where 0 ≤ γ ≤ 2 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e−|x|. We prove that for 0 ≤ γ < 1 the solutions develop blow-up in finite for a general class of initial data. In contrast we prove that for 1 < γ ≤ 2 the equation is globally well-posed.

[1]  M. Mimura,et al.  Pattern formation in interacting and diffusing systems in population biology. , 1982, Advances in biophysics.

[2]  Masayasu Mimura,et al.  Asymptotic Behavior for a Nonlinear Degenerate Diffusion Equation in Population Dynamics , 1983 .

[3]  R. D. Passo,et al.  Aggregative effects for a reaction-advection equation , 1984 .

[4]  T. Ikeda.  Stationary solutions of a spatially aggregating population model , 1984 .

[5]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[6]  T. Ikeda.  Standing pulse-like solutions of a spatially aggregating population model , 1985 .

[7]  Stability of localized stationary solutions , 1987 .

[8]  Masayasu Mimura,et al.  Localized cluster solutions of nonlinear degenerate diffusion equations arising in population dynamics , 1989 .

[9]  L. Rudin,et al.  Feature-oriented image enhancement using shock filters , 1990 .

[10]  L. Álvarez,et al.  Signal and image restoration using shock filters and anisotropic diffusion , 1994 .

[11]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[12]  Leah Edelstein-Keshet,et al.  Do travelling band solutions describe cohesive swarms? An investigation for migratory locusts , 1998 .

[13]  J. Toner,et al.  Flocks, herds, and schools: A quantitative theory of flocking , 1998, cond-mat/9804180.

[14]  A. Mogilner,et al.  A non-local model for a swarm , 1999 .

[15]  G. F.,et al.  From individuals to aggregations: the interplay between behavior and physics. , 1999, Journal of theoretical biology.

[16]  J. Hutchinson.  Animal groups in three dimensions , 1999 .

[17]  L. Edelstein-Keshet,et al.  Complexity, pattern, and evolutionary trade-offs in animal aggregation. , 1999, Science.

[18]  Dirk Helbing,et al.  Application of statistical mechanics to collective motion in biology , 1999 .

[19]  W Ebeling,et al.  Statistical mechanics of canonical-dissipative systems and applications to swarm dynamics. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  P. Constantin,et al.  On the critical dissipative quasi-geostrophic equation , 2001 .

[21]  W. Rappel,et al.  Self-organization in systems of self-propelled particles. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  A. Majda,et al.  Vorticity and incompressible flow , 2001 .

[23]  I. Couzin,et al.  Collective memory and spatial sorting in animal groups. , 2002, Journal of theoretical biology.

[24]  Werner Ebeling,et al.  Excitation of rotational modes in two-dimensional systems of driven Brownian particles. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Pierre Gilles Lemarié-Rieusset,et al.  Recent Developments in the Navier-Stokes Problem , 2002 .

[26]  Maximino Aldana,et al.  Phase Transitions in Self-Driven Many-Particle Systems and Related Non-Equilibrium Models: A Network Approach , 2003 .

[27]  A. Mogilner,et al.  Mathematical Biology Mutual Interactions, Potentials, and Individual Distance in a Social Aggregation , 2003 .

[28]  Werner Ebeling,et al.  COLLECTIVE MOTION OF BROWNIAN PARTICLES WITH HYDRODYNAMIC INTERACTIONS , 2003 .

[29]  Andrea L. Bertozzi,et al.  Swarming Patterns in a Two-Dimensional Kinematic Model for Biological Groups , 2004, SIAM J. Appl. Math..

[30]  A. Córdoba,et al.  Formation of singularities for a transport equation with nonlocal velocity , 2005, 0706.1969.

[31]  Marek Bodnar,et al.  Derivation of macroscopic equations for individual cell‐based models: a formal approach , 2005 .

[32]  A. Bertozzi,et al.  A Nonlocal Continuum Model for Biological Aggregation , 2005, Bulletin of mathematical biology.

[33]  M. Bodnar,et al.  An integro-differential equation arising as a limit of individual cell-based models , 2006 .

[34]  Andrea L. Bertozzi,et al.  Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn , 2007 .

[35]  Thomas Laurent,et al.  Local and Global Existence for an Aggregation Equation , 2007 .

[36]  Dong Li,et al.  Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation , 2008 .

[37]  Hongjie Dong,et al.  Finite time singularities for a class of generalized surface quasi-geostrophic equations , 2008 .

[38]  Martin Burger,et al.  Large time behavior of nonlocal aggregation models with nonlinear diffusion , 2008, Networks Heterog. Media.

[39]  Wellposedness and regularity of solutions of an aggregation equation , 2010 .

[40]  L. Edelstein-Keshet.  Mathematical models of swarming and social aggregation , .