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

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 $T$ as long as its initial date is a perturbation around the monotonic shear profile of small size like $e^{-T}$. 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

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

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

Optimal time-decay estimates for the compressible navier-stokes equations in the critical l p framework

The global existence issue for the isentropic compressible Navier-Stokes equations in the critical regularity framework has been addressed in [7] more than fifteen years ago. However, whether (optimal) time-decay rates could be shown in general critical spaces and any dimension d $\ge$ 2 has remained an open question. Here we give a positive answer to that issue not only in the L 2 critical framework of [7] but also in the more general L p critical framework of [3, 6, 14]. More precisely, we show that under a mild additional decay assumption that is satisfied if the low frequencies of the initial data are in e.g. L p/2 (R d), the L p norm (the slightly stronger $\dot B^0_{p,1}$ norm in fact) of the critical global solutions decays like t --d(1 p -- 1 4) for t $\rightarrow$ +$\infty$, exactly as firstly observed by A. Matsumura and T. Nishida in [23] in the case p = 2 and d = 3, for solutions with high Sobolev regularity. Our method relies on refined time weighted inequalities in the Fourier space, and is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics