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Register a new userKinetic equations with Maxwell boundary conditions
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Authors
Stephane Mischler
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We prove global stability results of {sl DiPerna-Lions} renormalized solutions for the initial boundary value problem associated to some kinetic equations, from which existence results classically follow. The (possibly nonlinear) boundary conditions are completely or partially diffuse, which includes the so-called Maxwell boundary conditions, and we prove that it is realized (it is not only a boundary inequality condition as it has been established in previous works). We are able to deal with Boltzmann, Vlasov-Poisson and Fokker-Planck type models. The proofs use some trace theorems of the kind previously introduced by the author for the Vlasov equations, new results concerning weak-weak convergence (the renormalized convergence and the biting $L^1$-weak convergence), as well as the Darroz`es-Guiraud information in a crucial way.
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We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(Omega)$ and $L_1(Omega)$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.
This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and S. Nicaise, Tech. Rep. 10-09, IRMAR (2010)] for the case with homogeneous boundary condition. Unfortunately, their method can not be applied to the inhomogeneous case. The main novelty of this paper is that we follow the spirit of the $H^1$-estimate in [R. Hiptmair, A. Moiola and I. Perugia, Math. Models Methods Appl. Sci., 21(2011), pp. 2263-2287] and modify the proof by applying two inequalities of Friedrichs type to make the $H^1$-estimate move into $H^2$-estimate and $W^{m, p}$-estimate.Finally, the dependence of the regularity estimates on the wave number is obtained, which will play an important role in the convergence analysis of the numerical solutions for the time-harmonic Maxwell equations.
We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $epsilon^{3/8 - delta}$ near the boundary and $epsilon^{3/4 - delta}$ in the interior, where $delta, delta$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.
We prove the existence of an Inertial Manifold for 3D complex Ginzburg-Landau equation with periodic boundary conditions as well as for more general cross-diffusion system assuming that the dispersive exponent is not vanishing. The result is obtained under the assumption that the parameters of the equation is chosen in such a way that the finite-time blow up of smooth solutions does not take place. For the proof of this result we utilize the recently suggested method of spatio-temporal averaging.
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