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Register a new userOn well-posedness of parabolic equations of Navier-Stokes type with BMO^{-1}(R^n) data
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We develop a strategy making extensive use of tent spaces to study parabolic equa-tions with quadratic nonlinearities as for the Navier-Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in R^n with small initial data in BMO^{-1}(R^n). We then study another model where neither pointwise kernel bounds nor self-adjointness are available.
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We give a new proof of a well-known result of Koch and Tataru on the well-posedness of Navier-Stokes equations in $R^n$ with small initial data in $BMO^{-1}(R^n)$. The proof is formulated operator theoretically and does not make use of self-adjointness of the Laplacian.
Existence and uniqueness of solutions to the Navier-Stokes equation in dimension two with forces in the space $L^q( (0,T); mathbf{W}^{-1,p}(Omega))$ for $p$ and $q$ in appropriate parameter ranges are proven. The case of spatially measured-valued inhomogeneities is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions with $1 < p, q < infty$ arbitrary.
In this paper, the initial-boundary value problem of the 1D full compressible Navier-Stokes equations with positive constant viscosity but with zero heat conductivity is considered. Global well-posedness is established for any $H^1$ initial data. The initial density is required to be nonnegative, which is not necessary to be uniformly away from vacuum. This not only generalizes the well-known result of Kazhikhov--Shelukhin (Kazhikhov, A.~V.; Shelukhin, V.~V.: emph{Unique global solution with respect to time of initial boundary value problems for one-dimensional equations of a viscous gas}, J.,Appl.,Math.,Mech., bf41 rm(1977), 273--282.) from the heat conductive case to the non-heat conductive case, and the initial vacuum is allowed.
In this paper, we consider the almost sure well-posedness of the Cauchy problem to the Cahn-Hilliard-Navier-Stokes equation with a randomization initial data on a torus $mathbb{T}^3$. First, we prove the local existence and uniqueness of solution. Furthermore, we prove the global existence and uniqueness of solution and give the relative probability estimate under the condition of small initial data.
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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