# American Institute of Mathematical Sciences

ISSN:
1937-5093

eISSN:
1937-5077

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## Kinetic & Related Models

June 2015 , Volume 8 , Issue 2

Issue on mathematical study on liquid crystals and related topics: Statics and dynamics

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2015, 8(2): 201-214 doi: 10.3934/krm.2015.8.201 +[Abstract](750) +[PDF](390.4KB)
Abstract:
We obtain new a priori estimates for spatially inhomogeneous solutions of a kinetic equation for granular media, as first proposed in [3] and, more recently, studied in [1]. In particular, we show that a family of convex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour of solutions. Furthermore, using an additional assumption on the interaction kernel and a potential for interaction'', we prove a global entropy estimate in the one-dimensional case.
2015, 8(2): 215-234 doi: 10.3934/krm.2015.8.215 +[Abstract](1368) +[PDF](436.4KB)
Abstract:
We present and analyze different splitting algorithms for numerical solution of the both classical and generalized nonlinear Schrödinger equations describing propagation of wave packets with special emphasis on applications to nonlinear fiber-optics. The considered generalizations take into account the higher-order corrections of the linear differential dispersion operator as well as the saturation of nonlinearity and the self-steepening of the field envelope function. For stabilization of the pseudo-spectral splitting schemes for generalized Schrödinger equations a regularization based on the approximation of the derivatives by the low number of Fourier modes is proposed. To illustrate the theoretically predicted performance of these schemes several numerical experiments have been done. In particular, we compute real-world examples of extreme pulses propagating in silica fibers.
2015, 8(2): 235-254 doi: 10.3934/krm.2015.8.235 +[Abstract](796) +[PDF](2838.2KB)
Abstract:
A kinetic model for the study of capillary flows in devices with microscale geometry is presented. The model is based on the Enskog-Vlasov kinetic equation and provides a reasonable description of both fluid-fluid and fluid-wall interactions. Numerical solutions are obtained by an extension of the classical Direct Simulation Monte Carlo (DSMC) to dense fluids. The equilibrium properties of liquid menisci between two hydrophilic walls are investigated and the validity of the Laplace-Kelvin equation at the microscale is assessed. The dynamical process which leads to the meniscus breakage is clarified.
2015, 8(2): 255-280 doi: 10.3934/krm.2015.8.255 +[Abstract](729) +[PDF](556.9KB)
Abstract:
In this paper we use the theory of viscosity solutions for Hamilton-Jacobi equations to study propagation phenomena in kinetic equations. We perform the hydrodynamic limit of some kinetic models thanks to an adapted WKB ansatz. Our models describe particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The scattering operator is supposed to satisfy a maximum principle. When the velocity space is bounded, we show, under suitable hypotheses, that the phase converges towards the viscosity solution of some constrained Hamilton-Jacobi equation which effective Hamiltonian is obtained solving a suitable eigenvalue problem in the velocity space. In the case of unbounded velocities, the non-solvability of the spectral problem can lead to different behavior. In particular, a front acceleration phenomena can occur. Nevertheless, we expect that when the spectral problem is solvable one can extend the convergence result.
2015, 8(2): 281-308 doi: 10.3934/krm.2015.8.281 +[Abstract](659) +[PDF](527.0KB)
Abstract:
We prove the immediate appearance of an exponential lower bound, uniform in time and space, for continuous mild solutions to the full Boltzmann equation in a $C^2$ convex bounded domain with the physical Maxwellian diffusion boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, with and without Grad's angular cutoff assumption. In particular, the lower bound is proven to be Maxwellian in the case of cutoff collision kernels. Moreover, these results are entirely constructive if the initial distribution contains no vacuum, with explicit constants depending only on the a priori bounds on the solution.
2015, 8(2): 309-333 doi: 10.3934/krm.2015.8.309 +[Abstract](717) +[PDF](515.2KB)
Abstract:
We consider the well-posedness problem for the space-homogeneous Boltzmann equation with soft-potential collision kernels. By revisiting the classical Fourier inequalities and fractional integrals, we deduce a set of bilinear estimates for the collision operator on the space of integrable functions possessing certain degree of smoothness and we apply them to prove the local-in-time existence of a solution to the Boltzmann equation in both integral form and the original one. Uniqueness and stability of solutions are also established.
2015, 8(2): 335-358 doi: 10.3934/krm.2015.8.335 +[Abstract](920) +[PDF](479.2KB)
Abstract:
We investigate the global existence and large-time behavior of classical solutions to the compressible Euler equations coupled to the incompressible Navier-Stokes equations. The coupled hydrodynamic equations are rigorously derived in [1] as the hydrodynamic limit of the Vlasov/incompressible Navier-Stokes system with strong noise and local alignment. We prove the existence and uniqueness of global classical solutions of the coupled system under suitable assumptions. As a direct consequence of our result, we can conclude that the estimates of hydrodynamic limit studied in [1] hold for all time. For the large-time behavior of the classical solutions, we show that two fluid velocities will be aligned with each other exponentially fast as time evolves.
2015, 8(2): 359-380 doi: 10.3934/krm.2015.8.359 +[Abstract](1105) +[PDF](531.8KB)
Abstract:
In this paper, we propose a kinetic model describing the collective motion by chemotaxis of two species in interaction emitting the same chemoattractant. Such model can be seen as a generalisation to several species of the Othmer-Dunbar-Alt model which takes into account the run-and-tumble process of bacteria. Existence of weak solutions for this two-species kinetic model is studied and the convergence of its diffusive limit towards a macroscopic model of Keller-Segel type is analysed.
2015, 8(2): 381-394 doi: 10.3934/krm.2015.8.381 +[Abstract](595) +[PDF](355.3KB)
Abstract:
We consider a self-gravitating collisionless gas where the gravitational interaction is modeled according to MOND (modified Newtonian dynamics). For the resulting modified Vlasov-Poisson system we establish the existence of spherically symmetric equilibria with compact support and finite mass. In the standard situation where gravity is modeled by Newton's law the latter properties only hold under suitable restrictions on the prescribed microscopic equation of state. Under the MOND regime no such restrictions are needed.

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