We develop a new approach to study the well-posedness theory of Prandtl equation in Sobolev spaces by using direct energy method under monotonicity condition on tangential velocity field instead Crocco transformation. Precisely, we firstly investigate linearized some weighted when background state is monotonic normal variable. Then cope with loss regularity perturbation respect due degeneracy equation, apply Nash-Moser-Hormander iteration obtain classical solutions nonlinear initial data...

In this paper we study the evolutions of interfaces betweengases and vacuum for both inviscid viscous one dimensionalisentropic gas motions. The local (in time) existence solutionsfor models with initial data containingvacuum states is proved some singular properties on freesurfaces separating are obtained. It isfound that Euler equations better behaved near thanthe compressible Navier-Stokes equations. Navier-Stokesequations viscosity depending density introduced, which shown to be...

For the viscous and heat-conductive fluids governed by compressible Navier–Stokes equations with an external potential force, there exist non-trivial stationary solutions zero velocity. By combining L p - q estimates for linearized elaborate energy method, convergence rates are obtained in various norms solution to profile whole space when initial perturbation of force small some Sobolev norms. More precisely, optimal its first order derivatives 2 -norm 1 is bounded.

In this paper, we study the one-dimensional motion of viscous gas connecting to vacuum state with a jump in density when viscosity depends on density. Precisely, coefficient μ is proportional ρθ and 0 < θ 1/2, where ρ density, global existence uniqueness weak solutions are proved. This improves previous results by enlarging interval θ.

In this paper, we are concerned with the optimal Lp–Lq convergence rates for compressible Navier–Stokes equations a potential external force in whole space. Under smallness assumption on both initial perturbation and some Sobolev spaces, of solution Lq-norm 2⩽q⩽6 its first order derivative L2-norm obtained when is bounded Lp 1⩽p<6/5. The proof based energy estimates to nonlinear problem semigroup generated by corresponding linearized operator.

Abstract We study the well‐posedness theory for MHD boundary layer. The layer equations are governed by Prandtl‐type that derived from incompressible system with non‐slip condition on velocity and perfectly conducting magnetic field. Under assumption initial tangential field is not zero, we establish local‐i‐time existence, uniqueness of solutions nonlinear equations. Compared classical Prandtl which monotonicity plays a crucial role, this needed This justifies physical understanding has...

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimensional compressible Navier--Stokes equations. Assume that corresponding Riemann problem Euler equations can be solved by (VR, UR, SR)(t,x). If initial data (v0 , u0 ,s0 )(x) nonisentropic a small perturbation an approximate wave constructed as in [S. Kawashima, A. Matsumura, and K. Nishihara, Proc. Japan Acad. Ser. A, 62 (1986), pp. 249--252], then we show that, for general gas,...

In this paper, we study a one-dimensional motion of viscous gas near vacuum with (or without) gravity. We are interested in the case that is contact at finite interval. This free boundary problem for isentropic Navier--Stokes equations, and boundaries interfaces separating from vacuum, across which density changes continuously. The regularity behavior solutions expanding rate studied. Smoothness discussed. uniqueness weak to also proved.

Most of the work on Boltzmann equation is based Grad'sangular cutoff assumption. Even though smoothing effect from thesingular cross-section without angular corresponding tothe grazing collision expected, there no general mathematicaltheory especially for spatially inhomogeneous case. As a furtherstudy problem in homogeneous situation, thispaper, we will prove Gevrey property solutionsto Cauchy Maxwellian molecules angularcutoff by using pseudo-differential calculus. Furthermore,...

In this paper, we study the global solutions with large data away from vacuum to Cauchy problem of one-dimensional compressible Navier--Stokes--Poisson system density-dependent viscosity coefficient and density- temperature-dependent heat-conductivity coefficient. The proof is based on some detailed analysis bounds density temperature functions.

The paper aims to justify the high Reynolds numbers limit for magnetohydrodynamics system with Prandtl boundary layer expansion when no-slip condition is imposed on a velocity field and perfectly conducting wall magnetic field. Under assumption that viscosity resistivity coefficients are of same order initial tangential not degenerate, we validity give an $L^\infty$ estimate error by multiscale analysis.

We study the well-posedness of Prandtl system without monotonicity and analyticity assumption. Precisely, for any index \sigma\in[3/2, 2], we obtain local in time space Gevrey class G^\sigma tangential variable Sobolev normal so that condition on velocity is not needed to overcome loss derivative. This answers open question raised by D. Gérard-Varet N. Masmoudi [Ann. Sci. École Norm. Sup. (4) 48 (2015), no. 6, 1273–1325], who solved case \sigma=7/4 .