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On the Cauchy problem for Boltzmann equations: global existence and weak stability
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In this article, the authors studied the large-data Cauchy problem for Boltzmann equations with general collision kernels and proved that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution.Abstract:
We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution. From this stability result we deduce global existence of a solution to the Cauchy problem. Our method relies upon recent compactness results for velocity averages, a new formulation of the Boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions. It allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations.read more
Citations
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Ordinary differential equations, transport theory and Sobolev spaces.
TL;DR: In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.
Journal ArticleDOI
Fluid dynamic limits of kinetic equations. I. Formal derivations
TL;DR: In this article, the incompressible Navier-Stokes equations were derived from a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of Hilbert or Chapman-Enskog.
Journal ArticleDOI
H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations
TL;DR: In this article, the H-means are introduced for studying oscillations and concentration effects in partial differential equations, and applications to transport properties and homogenisation are given as an example of the new results which can be obtained by this approach.
Journal ArticleDOI
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation
TL;DR: In this article, the authors derived estimates on the rate of convergence to equilibrium for solutions of the Boltzmann equation, like O(t-∞), which hold conditionally to some strong but natural estimates of smoothness, decay at large velocities and strict positivity.
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On a New Class of Weak Solutions to the Spatially Homogeneous Boltzmann and Landau Equations
TL;DR: In this article, the spatially homogeneous Boltzmannian equation without cut-off, the Fokker-Planck Landau equation, and the asymptotics of grazing collisions for a broad class of potentials were derived.
References
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Book
The Boltzmann equation and its applications
TL;DR: In this article, the Boltzmann Equation for rigid spheres is used to model the dynamics of a gas of rigid spheres in phase space and to solve the problem of flow and heat transfer in regions bounded by planes or cylinders.
Journal ArticleDOI
Ordinary differential equations, transport theory and Sobolev spaces.
TL;DR: In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.