# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

December 2016 , Volume 9 , Issue 4

Issue on Nonautonomous Dynamics

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2016, 9(4): 621-656 doi: 10.3934/krm.2016010 +[Abstract](1017) +[PDF](1543.1KB)
Abstract:
The mathematical description of the interaction between a plasma and a solid surface is a major issue that still remains challenging. In this paper, we model this interaction as a stationary and bi-kinetic Vlasov-Poisson-Ampère boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be determined from the ampibolarity of the particle flows as the unique solution of a non linear equation. Based on variational techniques, our analysis establishes the well-posedness of the model, provided that the incoming ion distribution satisfies a moment condition that generalizes the historical Bohm criterion of plasma physics. Quantitative estimates are also given, together with numerical illustrations that validate the robustness of our approach.
2016, 9(4): 657-686 doi: 10.3934/krm.2016011 +[Abstract](1154) +[PDF](490.7KB)
Abstract:
We study the time evolution of a single species positive plasma, confined in a cylinder and having infinite charge. We extend the result of a previous work by the same authors, for a plasma density having compact support in the velocities, to the case of a density having unbounded support and gaussian decay in the velocities.
2016, 9(4): 687-714 doi: 10.3934/krm.2016012 +[Abstract](1146) +[PDF](519.0KB)
Abstract:
This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < \frac{3d}{d+1}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(3-p)}{p}}$ norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.
2016, 9(4): 715-748 doi: 10.3934/krm.2016013 +[Abstract](1071) +[PDF](644.2KB)
Abstract:
This paper investigates the generalized Keller-Segel (KS) system with a nonlocal diffusion term $-\nu(-\Delta)^{\frac{\alpha}{2}}\rho~(1<\alpha<2)$. Firstly, the global existence of weak solutions is proved for the initial density $\rho_0\in L^1\cap L^{\frac{d}{\alpha}}(\mathbb{R}^d)~(d\geq2)$ with $\|\rho_0\|_{\frac {d}{\alpha}} < K$, where $K$ is a universal constant only depending on $d,\alpha,\nu$. Moreover, the conservation of mass holds true and the weak solution satisfies some hyper-contractive and decay estimates in $L^r$ for any $1< r<\infty$. Secondly, for the more general initial data $\rho_0\in L^1\cap L^2(\mathbb{R}^d)$$~(d=2,3)$, the local existence is obtained. Thirdly, for $\rho_0\in L^1\big(\mathbb{R}^d,(1+|x|)dx\big)\cap L^\infty(\mathbb{R}^d)(~d\geq2)$ with $\|\rho_0\|_{\frac{d}{\alpha}} < K$, we prove the uniqueness and stability of weak solutions under Wasserstein metric through the method of associating the KS equation with a self-consistent stochastic process driven by the rotationally invariant $\alpha$-stable Lévy process $L_{\alpha}(t)$. Also, we prove the weak solution is $L^\infty$ bounded uniformly in time. Lastly, we consider the $N$-particle interacting system with the Lévy process $L_{\alpha}(t)$ and the Newtonian potential aggregation and prove that the expectation of collision time between particles is below a universal constant if the moment $\int_{\mathbb{R}^d}|x|^\gamma\rho_0dx$ for some $1<\gamma<\alpha$ is below a universal constant $K_\gamma$ and $\nu$ is also below a universal constant. Meanwhile, we prove the propagation of chaos as $N\rightarrow\infty$ for the interacting particle system with a cut-off parameter $\varepsilon\sim(\ln N)^{-\frac{1}{d}}$, and show that the mean field limit equation is exactly the generalized KS equation.
2016, 9(4): 749-766 doi: 10.3934/krm.2016014 +[Abstract](956) +[PDF](439.5KB)
Abstract:
We study a modified Kac model where the classical kinetic energy is replaced by an arbitrary energy function $\phi(v)$, $v \in \mathbb{R}$. The aim of this paper is to show that the uniform density with respect to the microcanonical measure is $Ce^{-z_0\phi(v)}$-chaotic, $C,z_0 \in \mathbb{R}_+$. The kinetic energy for relativistic particles is a special case. A generalization to the case $v\in \mathbb{R}^d$ which involves conservation momentum is also formally discussed.
2016, 9(4): 767-776 doi: 10.3934/krm.2016015 +[Abstract](1095) +[PDF](375.7KB)
Abstract:
We consider the global existence of the two-dimensional Navier-Stokes flow in the exterior of a moving or rotating obstacle. Bogovski$\check{i}$ operator on a subset of $\mathbb{R}^2$ is used in this paper. One important thing is to show that the solution of the equations does not blow up in finite time in the sense of some $L^2$ norm. We also obtain the global existence for the 2D Navier-Stokes equations with linearly growing initial velocity.
2016, 9(4): 777-796 doi: 10.3934/krm.2016016 +[Abstract](927) +[PDF](464.0KB)
Abstract:
We establish a blow up criterion for the two-dimensional $k$-$\varepsilon$ model equations for turbulent flows in a bounded smooth domain $\Omega$. It is shown that for the initial-boundary value problem of the 2D $k$-$\varepsilon$ model equations in a bounded smooth domain, if $\|\nabla u\|_{L^{1}(0, T; L^{\infty})}+\|\nabla\rho\|_{L^{2}(0, T; L^{\infty})} +\|\varepsilon\|_{L^{2}(0, T; L^{\infty})}$ $<\infty$, then the strong solution $(\rho, u, h,k, \varepsilon)$ can be extended beyond $T$.
2016, 9(4): 797-797 doi: 10.3934/krm.2016017 +[Abstract](750) +[PDF](118.1KB)
Abstract:
The article is retracted by a unanimous decision by the three Editors-in-chief of KRM on the ground that almost the same article was submitted and got quickly published in Boundary Value Problems (DOI 10.1186/s13661-015-0481-7) by the same authors right before their submission to KRM.
2016, 9(4): 813-813 doi: 10.3934/krm.2016018 +[Abstract](895) +[PDF](189.4KB)
Abstract:
N/A

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