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Register a new userIll-posedness of Naiver-Stokes equations and critical Besov-Morrey spaces
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The blow up phenomenon in the first step of the classical Picards scheme was proved in this paper. For certain initial spaces, Bourgain-Pavlovic and Yoneda proved the ill-posedness of the Navier-Stokes equations by showing the norm inflation in certain solution spaces. But Chemin and Gallagher said the space $dot{B}^{-1,infty}_{infty}$ seems to be optimal for some solution spaces best chosen. In this paper, we consider more general initial spaces than Bourgain-Pavlovic and Yoneda did and establish ill-posedness result independent of the choice of solution space. Our result is a complement of the previous ill-posedness results on Navier-Stokes equations.
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In the paper, by constructing a initial data $u_{0}in B^{sigma}_{p,infty}$ with $sigma-2>max{1+frac 1 p, frac 3 2}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_{0}$ is discontinuous at $t=0$ in the norm of $B^{sigma}_{p,infty}$, which implies that the ill-posedness for the higher dimensional Camassa-Holm equations in $B^{sigma}_{p,infty}$.
We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in $mathbb{R}^n.$ We focus on the so-called critical Besov regularity framework. In this setting, it is natural to consider initial densities $rho_0,$ velocity fields $u_0$ and temperatures $theta_0$ with $a_0:=rho_0-1indot B^{frac np}_{p,1},$ $u_0indot B^{frac np-1}_{p,1}$ and $theta_0indot B^{frac np-2}_{p,1}.$ After recasting the whole system in Lagrangian coordinates, and working with the emph{total energy along the flow} rather than with the temperature, we discover that the system may be solved by means of Banach fixed point theorem in a critical functional framework whenever the space dimension is $ngeq2,$ and $1<p<2n.$ Back to Eulerian coordinates, this allows to improve the range of $p$s for which the system is locally well-posed, compared to Danchin, Comm. Partial Differential Equations 26 (2001).
We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banachs fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the $L^2(mathbb{R}^d)$-framework.
The compressible Navier-Stokes-Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions.
We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper particularly show an interesting phenomena on the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Youngs inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz-Leray projection, and the boundedness of the Riesz tranfroms are developed in mixed-norm Lebesgue spaces. These fundamental analysis results are independently topics of great interests and they are potentially useful in other problems.
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