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Kinetic & Related Models

2012 , Volume 5 , Issue 3

Issue dedicated to Michel Chipot
on the occasion of his 60th birthday

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A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem
Yong-Kum Cho
2012, 5(3): 441-458 doi: 10.3934/krm.2012.5.441 +[Abstract](197) +[PDF](398.8KB)
Abstract:
We present a Fourier transform formula of quadratic-form type for the collision operator with a Maxwellian kernel under the momentum transfer condition. As an application, we extend the work of Toscani and Villani on the uniform stability of the Cauchy problem for the associated Boltzmann equation to any physically relevant Maxwellian molecules in the long-range interactions with a minimal requirement for the initial data.
Hard sphere dynamics and the Enskog equation
Viktor I. Gerasimenko and Igor V. Gapyak
2012, 5(3): 459-484 doi: 10.3934/krm.2012.5.459 +[Abstract](293) +[PDF](484.0KB)
Abstract:
We develop a rigorous formalism for the description of the kinetic evolution of infinitely many hard spheres. On the basis of the kinetic cluster expansions of cumulants of groups of operators of finitely many hard spheres which are the generating operators of a nonperturbative solution of the Cauchy problem of the BBGKY hierarchy the nonlinear kinetic Enskog equation is derived. It is established that for initial states which are specified in terms of one-particle distribution functions the description of the evolution by the Cauchy problem of the BBGKY hierarchy and by the Cauchy problem of the generalized Enskog kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation are an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.
Optimization of a model Fokker-Planck equation
Michael Herty, Christian Jörres and Albert N. Sandjo
2012, 5(3): 485-503 doi: 10.3934/krm.2012.5.485 +[Abstract](303) +[PDF](439.1KB)
Abstract:
We discuss optimal control problems for the Fokker--Planck equation arising in radiotherapy treatment planning. We prove existence and uniqueness of an optimal boundary control for a general tracking--type cost functional in three spatial dimensions. Under additional regularity assumptions we prove existence of a continuous necessary first--order optimality system. In the one--dimensional case we analyse a numerical discretization of the Fokker--Planck equation. We prove that the resulting discrete optimality system is a suitable discretization of the continuous first--order system.
Regularity criteria for the 3D MHD equations via partial derivatives
Xuanji Jia and Yong Zhou
2012, 5(3): 505-516 doi: 10.3934/krm.2012.5.505 +[Abstract](277) +[PDF](362.6KB)
Abstract:
In this paper, we establish two regularity criteria for the 3D MHD equations in terms of partial derivatives of the velocity field or the pressure. It is proved that if $\partial_3 u \in L^\beta(0,T; L^\alpha(\mathbb{R}^3)),~\mbox{with}~ \frac{2}{\beta}+\frac{3}{\alpha}\leq\frac{3(\alpha+2)}{4\alpha},~\alpha>2$, or $\nabla_h P \in L^\beta(0,T; L^{\alpha}(\mathbb{R}^3)),~\mbox{with}~\frac{2}{\beta}+\frac{3}{\alpha}< 3,~\alpha>\frac{9}{7},~\beta\geq 1$, then the weak solution $(u,b)$ is regular on $[0, T]$.
On viscous quantum hydrodynamics associated with nonlinear Schrödinger-Doebner-Goldin models
José Luis López and Jesús Montejo-Gámez
2012, 5(3): 517-536 doi: 10.3934/krm.2012.5.517 +[Abstract](233) +[PDF](442.0KB)
Abstract:
The aim of this paper is to derive the quantum hydrodynamic system associated with the most general class of nonlinear Schrödinger equations accounting for Fokker--Planck type diffusion of the probability density, called of Doebner--Goldin. This 'Doebner--Goldin hydrodynamic system' is shown to be reduced in most cases to a simpler one of quantum Euler type by means of the introduction of a nonlinear gauge transformation that changes the fluid mean velocity into a new effective velocity corrected by an osmotic contribution. Finally, we also discuss some particular situations of especial interest and compare the structure of the resulting fluid systems with that of the viscous quantum hydrodynamic and the quantum Navier--Stokes equations stemming from maximization of the quantum entropy for Wigner--BGK models.
Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain
Ming Mei, Bruno Rubino and Rosella Sampalmieri
2012, 5(3): 537-550 doi: 10.3934/krm.2012.5.537 +[Abstract](243) +[PDF](387.8KB)
Abstract:
In this paper we present a physically relevant hydrodynamic model for a bipolar semiconductor device considering Ohmic conductor boundary conditions and a non-flat doping profile. For such an Euler-Poisson system, we prove, by means of a technical energy method, that the solutions are unique, exist globally and asymptotically converge to the corresponding stationary solutions. An exponential decay rate is also derived. Moreover we allow that the two pressure functions can be different.
A remark on Cannone-Karch solutions to the homogeneous Boltzmann equation for Maxwellian molecules
Yoshinori Morimoto
2012, 5(3): 551-561 doi: 10.3934/krm.2012.5.551 +[Abstract](242) +[PDF](275.5KB)
Abstract:
The purpose of this paper is to extend the result concerning the existence and the uniqueness of infinite energy solutions, given by Cannone-Karch, of the Cauchy problem for the spatially homogeneous Boltzmann equation of Maxwellian molecules without Grad's angular cutoff assumption in the mild singularity case, to the strong singularity case. This extension follows from a simple observation of the symmetry on the unit sphere for the Bobylev formula which is the Fourier transform of the Boltzmann collision term.
Convergence rates of zero diffusion limit on large amplitude solution to a conservation laws arising in chemotaxis
Hongyun Peng, Lizhi Ruan and Changjiang Zhu
2012, 5(3): 563-581 doi: 10.3934/krm.2012.5.563 +[Abstract](215) +[PDF](429.3KB)
Abstract:
In this paper, we investigate large amplitude solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. For the Cauchy problem and initial-boundary value problem, the global unique solvability is proved based on the energy method. In particular, our main purpose is to investigate the convergence rates as the diffusion parameter $\varepsilon$ goes to zero. It is shown that the convergence rates in $L^\infty$-norm are of the order $O\left(\varepsilon\right)$ and $O(\varepsilon^{1/2})$ corresponding to the Cauchy problem and the initial-boundary value problem respectively.
Optimal time decay of the non cut-off Boltzmann equation in the whole space
Robert M. Strain
2012, 5(3): 583-613 doi: 10.3934/krm.2012.5.583 +[Abstract](351) +[PDF](627.6KB)
Abstract:
In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space $\mathbb{R}^n _x$ with $n≥3$ .We use the existence theory of global in time nearby Maxwellian solutions from [12,11].It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption [26,1]. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\frac{N}{2}+\frac{N}{2r}})$ in the $L^2_v$$(L^r_x)$-norm for any $2\leq r\leq \infty$.
Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$
Zhong Tan, Yong Wang and Xu Zhang
2012, 5(3): 615-638 doi: 10.3934/krm.2012.5.615 +[Abstract](339) +[PDF](473.0KB)
Abstract:
We are concerned with the long-time behavior of global strong solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$, where the electric field is governed by the self-consistent Poisson equation. When the regular initial perturbations belong to $H^{4}(\mathbb{R}^{3})\cap \dot{B}_{1,\infty}^{-s}(\mathbb{R}^{3})$ with $s\in [0,1]$, we show that the density and momentum of the system converge to their equilibrium state at the optimal $L^2$-rates $(1+t)^{-\frac{3}{4}-\frac{s}{2}}$ and $(1+t)^{-\frac{1}{4}-\frac{s}{2}}$ respectively, and the decay rate is still $(1+t)^{-\frac{3}{4}}$ for temperature which is proved to be not optimal.
Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation
Alexander Zlotnik and Ilya Zlotnik
2012, 5(3): 639-667 doi: 10.3934/krm.2012.5.639 +[Abstract](313) +[PDF](1136.3KB)
Abstract:
We consider the time-dependent 1D Schrödinger equation on the half-axis with variable coefficients becoming constant for large $x$. We study a two-level symmetric in time (i.e. the Crank-Nicolson) and any order finite element in space numerical method to solve it. The method is coupled to an approximate transparent boundary condition (TBC). We prove uniform in time stability with respect to initial data and a free term in two norms, under suitable conditions on an operator in the approximate TBC. We also consider the corresponding method on an infinite mesh on the half-axis. We derive explicitly the discrete TBC allowing us to restrict the latter method to a finite mesh. The operator in the discrete TBC is a discrete convolution in time; in turn its kernel is a multiple discrete convolution. The stability conditions are justified for it. The accomplished computations confirm that high order finite elements coupled to the discrete TBC are effective even in the case of highly oscillating solutions and discontinuous potentials.
Erratum to: Ghost effect by curvature in planar Couette flow [1]
Leif Arkeryd, Raffaele Esposito, Rossana Marra and Anne Nouri
2012, 5(3): 669-672 doi: 10.3934/krm.2012.5.669 +[Abstract](174) +[PDF](270.9KB)
Abstract:
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