# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

September 2015 , Volume 8 , Issue 3

Issue on the workshop “Electromagnetics-Modelling, Simulation, Control and Industrial Applications"

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2015, 8(3): 395-411 doi: 10.3934/krm.2015.8.395 +[Abstract](1789) +[PDF](433.0KB)
Abstract:
This paper is devoted to the study of the inviscid Boussinesq equations. We establish the local well-posedness and blow-up criteria in Besov-Morrey spaces $N_{p,q,r}^s(\mathbb{R}^n)$ for super critical case $s > 1 + \frac{n}{p}, 1 < q \leq p < \infty, 1 \leq r\leq \infty$, and critical case $s=1+\frac{n}{p}, 1 < q \leq p < \infty, r=1$. Main analysis tools are Littlewood-Paley decomposition and the paradifferential calculus.
2015, 8(3): 413-441 doi: 10.3934/krm.2015.8.413 +[Abstract](3040) +[PDF](6359.2KB)
Abstract:
The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient $\epsilon$ is varied from $\epsilon=1$ (for 1D transport models) to $\epsilon=0$ (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit $\epsilon\to 0$, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams.
2015, 8(3): 443-465 doi: 10.3934/krm.2015.8.443 +[Abstract](2638) +[PDF](441.4KB)
Abstract:
The rigorous derivation of the Uehling-Uhlenbeck equation from more fundamental quantum many-particle systems is a challenging open problem in mathematics. In this paper, we exam the weak coupling limit of quantum $N$ -particle dynamics. We assume the integral of the microscopic interaction is zero and we assume $W^{4,1}$ per-particle regularity on the coressponding BBGKY sequence so that we can rigorously commute limits and integrals. We prove that, if the BBGKY sequence does converge in some weak sense, then this weak-coupling limit must satisfy the infinite quantum Maxwell-Boltzmann hierarchy instead of the expected infinite Uehling-Uhlenbeck hierarchy, regardless of the statistics the particles obey. Our result indicates that, in order to derive the Uehling-Uhlenbeck equation, one must work with per-particle regularity bound below $W^{4,1}$.
2015, 8(3): 467-492 doi: 10.3934/krm.2015.8.467 +[Abstract](1791) +[PDF](556.3KB)
Abstract:
The aim of this paper is the rigorous derivation of a stochastic non-linear diffusion equation from a radiative transfer equation perturbed with a random noise. The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The Hilbert expansion has to be done up to order 3 to overcome some difficulties caused by the random noise.
2015, 8(3): 493-531 doi: 10.3934/krm.2015.8.493 +[Abstract](1636) +[PDF](562.7KB)
Abstract:
We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity.
2015, 8(3): 533-558 doi: 10.3934/krm.2015.8.533 +[Abstract](1692) +[PDF](370.2KB)
Abstract:
In plasma physics domain, the electrons transport can be described from kinetic and hydrodynamical models. Both methods present disadvantages and thus cannot be considered in practical computations for Inertial Confinement Fusion (ICF). That is why we propose in this paper a new model which is intermediate between these two descriptions. More precisely, the derivation of such models is based on an angular closure in the phase space and retains only the energy of particles as a kinetic variable. The closure of the moment system is obtained from a minimum entropy principle. The resulting continuous model is proved to satisfy fundamental properties. Moreover the model is discretized w.r.t the energy variable and the semi-discretized scheme is shown to satisfy conservation properties and entropy decay.
2015, 8(3): 559-585 doi: 10.3934/krm.2015.8.559 +[Abstract](1549) +[PDF](491.1KB)
Abstract:
In this paper, we consider the time-asymptotic stability of a superposition of shock waves with contact discontinuities for the one dimensional Jin-Xin relaxation system with small initial perturbations, provided that the strengths of waves are small with the same order. The results are obtained by elementary weighted energy estimates based on the underlying wave structure and a delicate decay estimate on the heat kernel in Huang-Li-Matsumura [5].
2015, 8(3): 587-613 doi: 10.3934/krm.2015.8.587 +[Abstract](2128) +[PDF](663.1KB)
Abstract:
We deal with the initial-boundary value problem for the 1D time-dependent Schrödinger equation on the half-axis. The finite-difference scheme with the Numerov averages on the non-uniform space mesh and of the Crank-Nicolson type in time is studied, with some approximate transparent boundary conditions (TBCs). Deriving bounds for the skew-Hermitian parts of the Numerov sesquilinear forms, we prove the uniform in time stability in $L^2$- and $H^1$-like space norms under suitable conditions on the potential and the meshes. In the case of the discrete TBC, we also derive higher order in space error estimates in both norms in dependence with the Sobolev regularity of the initial function (and the potential) and properties of the space mesh. Numerical results are presented for tunneling through smooth and rectangular potentials-wells, including the global Richardson extrapolation in time to ensure higher order in time as well.
2015, 8(3): 615-616 doi: 10.3934/krm.2015.8.615 +[Abstract](1505) +[PDF](226.5KB)
Abstract:
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