Some remarks on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem

The aim of the paper is to investigate on some questions of local regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. The results are obtained in the wake of the ones, well known, by Caffarelli-Kohn-Nirenberg

Global existence of solutions to 2-D Navier-Stokes flow with non-decaying initial data in half-plane

We investigate the Navier-Stokes initial boundary value problem in the half-plane $R^2_+$ with initial data $u_0 \in L^\infty(R^2_+)\cap J_0^2(R^2_+)$ or with non decaying initial data $u_0\in L^\infty(R^2_+) \cap J_0^p(R^2_+), p > 2$ . We introduce a technique that allows to solve the two-dimesional problem, further, but not least, it can be also employed to obtain weak solutions, as regards the non decaying initial data, to the three-dimensional Navier-Stokes IBVP. This last result is the first of its kind

An elementary proof of uniqueness of the particle trajectories for solutions of a class of shear-thinning non-Newtonian 2D fluids

We prove some regularity results for a class of two dimensional non-Newtonian fluids. By applying results from [Dashti and Robinson, Nonlinearity, 22 (2009), 735-746] we can then show uniqueness of particle trajectories

A new proof of existence in the L^3-setting of solutions to the Navier–Stokes Cauchy problem

We investigate the existence of solutions with initial datum U_0 in L^3. Our main goal is to establish the existence interval (0, T), uniquely considering the size ||U_0||_3 and the absolute continuity of |U_0(x)|^

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

Abstract: We get results of existence and of uniqueness of solutions assuming that at initial data has a suitable spatial behaviour. We study pointwise stability and asymptotic behaviour in (t,x) variables of the solutions. The region of the motion is the half space and the solutions can be with non finite kinetic energy. The difficulty is connected with the fact that the Green function is expressed by a convolution product between the heat kernel and the fundamental solutions of Laplace equation

A high regularity result of solutions to a modified p-Navier-Stokes system

In a previous paper we established a result of high regularity of solutions to a modified p-Stokes problem, p is an element of (1, 2). By this expression we mean a perturbed p-Laplacian system. Here we prove that for a suitable body force there exists at least a solution to a modified p-Navier-Stokes problem, whose regularity is "high". More precisely, without restrictions on the size of the body force, for p close to 2, we prove that there exist second derivatives which are integrable on the whole domain R^3. Of course, the interest of the result is connected to the fact that for the first time a result of high regularity is deduced for solutions to a system of p-Navier-Stokes kind. It is also interesting to point out that the proof, based on the results of the p-Stokes problem, seems to be original and applicable to other nonlinear equations

On the (x,t)(x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space.

Abstract: We get results of existence and of uniqueness of solutions assuming that at initial data has a suitable spatial behaviour. We study pointwise stability and asymptotic behaviour in (t,x) variables of the solutions. The region of the motion is the half space and the solutions can be with non finite kinetic energy. The difficulty is connected with the fact that the Green function is expressed by a convolution product between the heat kernel and the fundamental solutions of Laplace equation

SOME REMARKS ON THE PARTIAL REGULARITY OF A SUITABLE WEAK SOLUTION TO THE NAVIER–STOKES CAUCHY PROBLEM

The goal of the paper is to explore some of the issues related to local regularity of a suitable weak solution to the Navier–Stokes Cauchy problem. The obtained results are in the spirit of the well-known results by Caffarelli–Kohn–Nirenberg