15,836 research outputs found

    Incompressible Navier-Stokes-Fourier Limit from The Boltzmann Equation: Classical Solutions

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    The global classical solution to the incompressible Navier-Stokes-Fourier equation with small initial data in the whole space is constructed through a zero Knudsen number limit from the solutions to the Boltzmann equation with general collision kernels. The key point is the uniform estimate of the Sobolev norm on the global solutions to the Boltzmann equation.Comment: 21 page

    Existence and convexity of local solutions to degenerate hessian equations

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    In this work, we prove the existence of local convex solution to the degenerate Hessian equationComment: corrections some typos in this versio

    Ultra-analytic effect of Cauchy problem for a class of kinetic equations

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    The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous non linear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.Comment: 22 pages, to appear in JD

    Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation

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    In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section. We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time. This is a Gevrey regularizing effect for non smooth initial datum. The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier

    Long time well-posdness of the Prandtl equations in Sobolev space

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    In this paper, we study the long time well-posedness for the nonlinear Prandtl boundary layer equation on the half plane. While the initial data are small perturbations of some monotonic shear profile, we prove the existence, uniqueness and stability of solutions in weighted Sobolev space by energy methods. The key point is that the life span of the solution could be any large TT as long as its initial date is a perturbation around the monotonic shear profile of small size like e−Te^{-T}. The nonlinear cancellation properties of Prandtl equations under the monotonic assumption are the main ingredients to establish a new energy estimate.Comment: In this version, reviser some typos, 43 page

    C1,1C^{1, 1} Solution of the Dirichlet Problem for Degenerate kk-Hessian Equations

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    In this paper, we prove the existence of C1,1C^{1,1}-solution to the Dirichlet problem for degenerate elliptic kk-Hessian equations Sk[u]=fS_{k}[u]=f under a condition which is weaker than the condition f1/k∈C1,1(Ωˉ)f^{1/k}\in C^{1,1}(\bar\Omega).Comment: 18page
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