### Stabilizing influence of a Skew-symmetric operator in semilinear parabolic equation

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We assume that $\mathbb{v}$ is a weak solution to the non-steady Navier-Stokes initial-boundary value problem that satisfies the strong energy inequality in its domain and the Prodi-Serrin integrability condition in the neighborhood of the boundary. We show the consequences for the regularity of $\mathbb{v}$ near the boundary and the connection with the interior regularity of an associated pressure and the time derivative of $\mathbb{v}$.

The paper contains the proof of global existence of weak solutions to the mixed initial-boundary value problem for a certain modification of a system of equations of motion of viscous compressible fluid. The modification is based on an application of an operator of regularization to some terms appearing in the system of equations and it does not contradict the laws of fluid mechanics. It is assumed that pressure is a known function of density. The method of discretization in time is used and finally,...

We deal with a suitable weak solution $(\mathbf{v},p)$ to the Navier-Stokes equations in a domain $\Omega \subset {\mathbb{R}}^{3}$. We refine the criterion for the local regularity of this solution at the point $(\mathbf{f}{x}_{0},{t}_{0})$, which uses the ${L}^{3}$-norm of $\mathbf{v}$ and the ${L}^{3/2}$-norm of $p$ in a shrinking backward parabolic neighbourhood of $({\mathbf{x}}_{0},{t}_{0})$. The refinement consists in the fact that only the values of $\mathbf{v}$, respectively $p$, in the exterior of a space-time paraboloid with vertex at $({\mathbf{x}}_{0},{t}_{0})$, respectively in a ”small” subset of this exterior, are considered. The consequence is that...

It is shown that the uniform exponential stability and the uniform stability at permanently acting disturbances of a sufficiently smooth but not necessarily steady-state solution of a general variational inequality is a consequence of the uniform exponential stability of a zero solution of another (so called linearized) variational inequality.

We formulate a boundary value problem for the Navier-Stokes equations with prescribed u·n, curl u·n and alternatively (∂u/∂n)·n or curl²u·n on the boundary. We deal with the question of existence of a steady weak solution.

We formulate sufficient conditions for regularity up to the boundary of a weak solution v in a subdomain Ω × (t₁,t₂) of the time-space cylinder Ω × (0,T) by means of requirements on one of the eigenvalues of the rate of deformation tensor. We assume that Ω is a cube.

The concept of regularization to the complete system of Navier-Stokes equations for viscous compressible heat conductive fluid is developed. The existence of weak solutions for the initial boundary value problem for the modified equations is proved. Some energy and etropy estimates independent of the parameter of regularization are derived.

We study axisymmetric solutions to the Navier-Stokes equations in the whole three-dimensional space. We find conditions on the radial and angular components of the velocity field which are sufficient for proving the regularity of weak solutions.

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