Related papers: On global weak solutions to the Cauchy problem for…
In this short note we address a problem raised in [21], concerning the uniqueness of solutions to Naiver Stokes equation with small initial data in $L^{3,\infty}(R^3)$, the Lorentz space. We prove uniqueness for such initial data.
We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \begin{align} u_t+(-\Delta)^{\alpha}u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show the analyticity of…
We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a solution to a stochastic problem. Namely, we construct diffusion processes which allow us to obtain a probabilistic…
The initial value problem of the incompressible Navier-Stokes equations with non-zero forces in $L^{n,\infty}(\mathbb{R}^n)$ is investigated. Even though the Stokes semigroup is not strongly continuous on $L^{n,\infty}(\mathbb{R}^n)$, with…
We find a simple quantitative lower bound for lifespan of solution of the multidimensional initial value problem for the Navier-Stokes equations in whole space when the initial function belongs to the correspondent Lebesgue-Riesz space, and…
The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier-Stokes system. The Marcinkiewicz space $L^{3,\infty}$ is used to prove some asymptotic stability…
We consider the Navier-Stokes Cauchy problem with an initial datum in a weighted Lebesgue space. The weight is a radial function increasing at infinity. Our study partially follows the ideas of the paper by G.P. Galdi and P. Maremonti "On…
In this paper, we study the initial and boundary value problem of the Navier-Stokes equations in the half space. We prove the unique existence of weak solution $u\in L^q(\R_+\times (0,T))$ with $\nabla u\in L^{\frac{q}{2}}_{loc}(\R_+\times…
We prove global existence of solutions to the Cauchy problem for the compressible Navier-Stokes equations in Euclidean spaces, given initial data with small norms in Besov and critical weighted Besov spaces. Global existence and a priori…
We study the existence of a strong solution to the initial value problem for the incompressible Navier-Stokes equations in the whole space. Our investigation shows that a ``suitable'' weak solution to the problem becomes a strong one…
The weak solution to the Navier-Stokes equations in a bounded domain $D \subset \mathbb{R}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t \geq 0$. In a…
In this paper, we study the Cauchy problem to the 3D fractional compressible isentropic generalized Navier-Stokes equations for viscous compressible fluid with one Levy diffusion process. We obtain the existence and uniqueness of global…
A well-known unsolved problem (in the classical theory of fluid mechanics) is to identify a set of initial velocities, which may depend on the viscosity, the body forces and possibly the boundary of the fluid that will allow global in time…
In this paper we will prove that the vorticity belongs to L1(0; T ; L2(R3)) for the Cauchy problem of 3D incompressible Navier-Stokes equation, then the existence of a global smooth solution is obtained. Our approach is to construct a set…
In this paper, we investigate the existence of a unique global smooth solution to the three-dimensional incompressible Navier-Stokes equations and provide a concise proof. We establish a new global well-posedness result that allows the…
In 1934 Leray proved that the Navier-Stokes equations have global weak solutions for initial data in $L^2(\mathbb{R}^N)$. In 1990 Calder\'on extended this result to the initial value spaces $L^p(\mathbb{R}^N)$ ($2\leq p<\infty$). In the…
The paper is concerned with the IBVP of the Navier-Stokes equations. The goal is the construction of a weak solution enjoying some new properties. Of course, we look for properties which are global in time. The results hold assuming an…
We study local regularity properties of a weak solution $u$ to the Cauchy problem of the incompressible Navier-Stokes equations. We present a new regularity criterion for the weak solution $u$ satisfying the condition…
Some known results regarding the Euler and Navier-Stokes equations were obtained by different authors. Existence and smoothness of the Navier-Stokes solutions in two dimensions have been known for a long time. Leray $\cite{jL34}$ showed…
In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power ($\rho^\delta$ with…