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Kinetic & Related Models

2015 , Volume 8 , Issue 4

Issue on rate-independent evolutions and hysteresis modelling

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Some a priori estimates for the homogeneous Landau equation with soft potentials
Radjesvarane Alexandre, Jie Liao and Chunjin Lin
2015, 8(4): 617-650 doi: 10.3934/krm.2015.8.617 +[Abstract](67) +[PDF](592.7KB)
This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in $L^2$ space hold true for moderately soft potential cases $ \gamma \in[-2, 0) $ without any smallness assumption on the initial data. For very soft potential cases $ \gamma \in[-3, -2) $, which cover in particular the Coulomb case $\gamma=-3$, we get local in time estimates of weak solutions in $L^{2}$.
    In the proofs of these estimates, global ones for the special case $\gamma=-2$ and local ones for very soft potential cases $ \gamma \in[-3, -2) $, the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality.
Confinement by biased velocity jumps: Aggregation of escherichia coli
Vincent Calvez, Gaël Raoul and Christian Schmeiser
2015, 8(4): 651-666 doi: 10.3934/krm.2015.8.651 +[Abstract](64) +[PDF](667.1KB)
We investigate a one-dimensional linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint, it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.
A free boundary problem for a class of parabolic type chemotaxis model
Hua Chen, Wenbin Lv and Shaohua Wu
2015, 8(4): 667-684 doi: 10.3934/krm.2015.8.667 +[Abstract](61) +[PDF](392.7KB)
In this paper, we study a free boundary problem for a class of parabolic type chemotaxis model in high dimensional symmetry domain $\Omega$. By using the contraction mapping principle and operator semigroup approach, we establish the existence of the solution for such kind of chemotaxis system in the domain $\Omega$ with free boundary condition.
Strong continuity for the 2D Euler equations
Gianluca Crippa, Elizaveta Semenova and Stefano Spirito
2015, 8(4): 685-689 doi: 10.3934/krm.2015.8.685 +[Abstract](55) +[PDF](323.0KB)
We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second result provides uniform continuity and is restricted to Hölder continuous solutions.
Energy dissipation for weak solutions of incompressible liquid crystal flows
Shanshan Guo and Zhong Tan
2015, 8(4): 691-706 doi: 10.3934/krm.2015.8.691 +[Abstract](77) +[PDF](464.4KB)
In this paper, we are concerned with the simplified Ericksen-Leslie system (1)--(3), modeling the flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0, d_0)\in {\bf H}\times H^1(\Omega, \mathbb{S}^2)$, with $d_0(\Omega)\subset\mathbb{S}^2_+$. We define a dissipation term $D(u,d)$ that stems from an eventual lack of smoothness in the solutions, and then obtain a local equation of energy for weak solutions of liquid crystals in dimensions three. As a consequence, we consider the 2D case and obtain $D(u,d)=0$.
An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach
Jingwei Hu, Shi Jin and Li Wang
2015, 8(4): 707-723 doi: 10.3934/krm.2015.8.707 +[Abstract](59) +[PDF](727.0KB)
We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions --- a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since the equilibrium solution is a (zero-momentum) Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision [10] cannot capture the correct energy-transport limit. This problem was addressed in [13], where a thresholded BGK penalization was introduced. Here we propose an alternative based on a splitting approach. It has the advantage of treating the collisions at different scales separately, hence is free of choosing threshold and easier to implement. Formal asymptotic analysis and numerical results validate the efficiency and accuracy of the proposed scheme.
Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime
Thomas Leroy
2015, 8(4): 725-763 doi: 10.3934/krm.2015.8.725 +[Abstract](57) +[PDF](601.6KB)
We consider the relativistic transfer equations for photons interacting via emission absorption and scattering with a moving fluid. We prove a comparison principle and we study the non-equilibrium regime: the relativistic correction terms in the scattering operator lead to a frequency drift term modeling the Doppler effects. We prove that the solution of the relativistic transfer equations converges toward the solution of this drift diffusion equation.
Strong solutions to compressible barotropic viscoelastic flow with vacuum
Tong Tang and Yongfu Wang
2015, 8(4): 765-775 doi: 10.3934/krm.2015.8.765 +[Abstract](74) +[PDF](350.3KB)
We consider strong solutions to compressible barotropic viscoelastic flow in a domain $\Omega\subset\mathbb{R}^{3}$ and prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Inspired by the work of Kato and Lax, we use the contraction mapping principle to get the result.
Global existence and steady states of a two competing species Keller--Segel chemotaxis model
Qi Wang, Lu Zhang, Jingyue Yang and Jia Hu
2015, 8(4): 777-807 doi: 10.3934/krm.2015.8.777 +[Abstract](120) +[PDF](884.7KB)
We study a one--dimensional quasilinear system proposed by J. Tello and M. Winkler [27] which models the population dynamics of two competing species attracted by the same chemical. The kinetic terms of the interacting species are chosen to be of the Lotka--Volterra type and the boundary conditions are of homogeneous Neumann type which represent an enclosed domain. We prove the global existence and boundedness of classical solutions to the fully parabolic system. Then we establish the existence of nonconstant positive steady states through bifurcation theory. The stability or instability of the bifurcating solutions is investigated rigorously. Our results indicate that small intervals support stable monotone positive steady states and large intervals support nonmonotone steady states. Finally, we perform extensive numerical studies to demonstrate and verify our theoretical results. Our numerical simulations also illustrate the formation of stable steady states and time--periodic solutions with various interesting spatial structures.

2016  Impact Factor: 1.261




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