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Philippe Laurençot

Publications and source records attributed to Philippe Laurençot.

99 records · Page 6Linked to original sources
arXiv2008

Fermi-Dirac-Fokker-Planck equation: well-posedness and long-time asymptotics

A Fokker-Planck type equation for interacting particles with exclusion principle is analysed. The nonlinear drift gives rise to mathematical difficulties in controlling moments of the distribution function. Assuming enough initial moments are finite, we can show the global existence of weak solutions for this problem. The natural associated entropy of the equation is the main tool to derive uniform in time a priori estimates for the kinetic energy and entropy. As a consequence, long-time asymptotics in $L^1$ are characterized by the Fermi-Dirac equilibrium with the same initial mass. This result is achieved without rate for any constructed global solution and with exponential rate due to entropy/entropy-dissipation arguments for initial data controlled by Fermi-Dirac distributions. Finally, initial data below radial solutions with suitable decay at infinity lead to solutions for which the relative entropy towards the Fermi-Dirac equilibrium is shown to converge to zero without decay rate.

math.AP↗
arXiv2008

Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with $d\ge3$ and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass $M_c$ such that if $M \in (0,M_c]$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for $M \in (0,M_c)$. While characterising the eventual infinite time blowing-up profile for $M=M_c$, we observe that the long time asymptotics are much more complicated than in the classical Patlak-Keller-Segel system in dimension two.

math.AP↗
arXiv2007

On an age and spatially structured population model for Proteus Mirabilis swarm-colony development

Proteus mirabilis are bacteria that make strikingly regular spatial-temporal patterns on agar surfaces. In this paper we investigate a mathematical model that has been shown to display these structures when solved numerically. The model consists of an ordinary differential equation coupled with a partial differential equation involving a first-order hyperbolic aging term together with nonlinear degenerate diffusion. The system is shown to admit global weak solutions.

q-bio.PE↗
arXiv2007

Localized non-diffusive asymptotic patterns for nonlinear parabolic equations with gradient absorption

We study the large-time behaviour of the solutions $u$ of the evolution equation involving nonlinear diffusion and gradient absorption $\partial_t u - Δ_p u + |\nabla u|^q=0$. We consider the problem posed for $x\in {\mathbb R}^N $ and $t>0$ with non-negative and compactly supported initial data. We take the exponent $p>2$ which corresponds to slow $p$-Laplacian diffusion, and the exponent $q$ in the superlinear range $1<q<p-1$. In this range the influence of the Hamilton-Jacobi term $ |\nabla u|^q$ is determinant, and gives rise to the phenomenon of localization. The large time behaviour is described in terms of a suitable self-similar solution that solves a Hamilton-Jacobi equation. The shape of the corresponding spatial pattern is rather conical instead of bell-shaped or parabolic.

math.AP↗
arXiv2007

Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as time goes to infinity of the mass of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established.

math.AP↗
arXiv2007

Asymptotic profiles of solutions to convection-diffusion equations

The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by the derivative of the Guass-Weierstrass kernel or by a self-similar solution or by a hyperbolic N-wave

math.AP↗
arXiv2006

Asymptotic behavior for a viscous Hamilton-Jacobi equation with critical exponent

The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - Δu + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a rescaled self-similar solution of the linear heat equation is shown, the rescaling factor being $(\log(t))^{-(N+1)}$. The proof relies on the construction of a one-dimensional invariant manifold for a suitable truncation of the equation written in self-similar variables.

math.AP↗
arXiv2006

Decay Estimates for a Viscous Hamilton-Jacobi Equation with Homogenious Dirichet Boundary Conditions

Global classical solutions to the viscous Hamilton-Jacobi equation with homogenious Dirichlet boundary conditions are shown to converge to zero at the same speed as the linear heat semigroup when p > 1. For p = 1, an exponential decay to zero is also obtained in one space dimension but the rate depends on a and differs from that of the linear heat equation. Finally, if 0 < p < 1 and a < 0, finite time extinction occurs for non-negative solutions.

math.AP↗