In this paper we study the validity of the so-called Oberbeck-Boussinesq approximation for compressible viscous perfect gases in the whole three-dimensional space. Both the cases of uids with positive heat conductivity and zero conductivity are considered. For small perturbations of a constant equilibrium, we establish the global existence of unique strong solutions in a critical regularity functional framework. Next, taking advantage of Strichartz estimates for the associated system of acoustic waves, and of uniform estimates with respect to the Mach number, we obtain all-time convergence to the Boussinesq system with a explicit decay rate.
There is an important literature dedicated to the limit system, that is the Oberbeck-Boussinesq equations , and , under various hypotheses over the coefficients @math and @math and the potential @math (although the most common assumption is that @math ). Loosely speaking the classical results concerning the existence issue are (see e.g. @cite_17 @cite_5 @cite_10 and the references therein): Dimension @math : Global existence of strong solutions if @math Dimension @math with @math : Global weak solutions and local strong solutions (which become global if the data are small). Dimension @math with @math : only local-in-time strong solutions are available.
@cite_5 @cite_10 @cite_17
2014137435 2094712512
This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in @math space hold true for moderately soft potential cases @math without any smallness assumption on the initial data. For very soft potential cases @math , which cover in particular the Coulomb case @math , we get local in time estimates of weak solutions in @math . In the proofs of these estimates, global ones for the special case @math and local ones for very soft potential cases @math , the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality. We seek a global minimum of @math . The solution to @math will find local minima. Using the idea of simulated annealing, we consider the diffusion process, @math , @math , where @math is the n-dimensional standard Brownian motion and @math is the annealing rate which decreases to zero as t goes to @math . Under suitable condition on @math , we prove that @math converges weakly to a probability measure @math if for large t, @math with @math , where @math has a simple expression involving the action function of the dynamical system @math , @math concentrates on the global minima of U and is the weak limit of the Gibbs densities @math .The above result can also be formulated as follows: consider the FokkerPlanck equation (forward equation) [ t V(t,y) = ... Wave maps are the simplest wave equations taking their values in a Riemannian manifold (M,g). Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric g. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as @math , present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of KlainermanMachedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for @math as target. This monograph establishes that for @math as target the wave map evolution of any smooth data exists globally as a smooth function. While we restrict ourselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations.