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### A direct proof of the Caffarelli-Kohn-Nirenberg theorem

Banach Center Publications

In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if $limsu{p}_{R\to 0⁺}1/R{\int }_{{Q}_{R}\left(x₀,t₀\right)}|curlu×u/|u||²dxdt\le {\epsilon }_{*}$ for a sufficiently small ${\epsilon }_{*}>0$.

### A geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations

Mathematica Bohemica

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

### A mathematical model for resin transfer molding

Annales mathématiques Blaise Pascal

### A new regularity criterion for strong solutions to the Ericksen-Leslie system

Applicationes Mathematicae

A regularity criterion for strong solutions of the Ericksen-Leslie equations is established in terms of both the pressure and orientation field in homogeneous multiplier spaces.

### A new regularity criterion for the Navier-Stokes equations.

The Journal of Nonlinear Sciences and its Applications

### A note on a degenerate elliptic equation with applications for lakes and seas.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### A note on the generalized energy inequality in the Navier-Stokes equations

Applications of Mathematics

We prove that there exists a suitable weak solution of the Navier-Stokes equation, which satisfies the generalized energy inequality for every nonnegative test function. This improves the famous result on existence of a suitable weak solution which satisfies this inequality for smooth nonnegative test functions with compact support in the space-time.

### A parabolic system involving a quadratic gradient term related to the Boussinesq approximation.

RACSAM

We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. Based on adequate notions of weak and strong...

### A RANS 3D model with unbounded eddy viscosities

Annales de l'I.H.P. Analyse non linéaire

### A regularity criterion for the Navier-Stokes equations in terms of the horizontal derivatives of the two velocity components.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### A regularity criterion for the Navier-Stokes equations in terms of the pressure gradient

Open Mathematics

The incompressible three-dimensional Navier-Stokes equations are considered. A new regularity criterion for weak solutions is established in terms of the pressure gradient.

### A remark on the regularity for the 3D Navier-Stokes equations in terms of the two components of the velocity.

Electronic Journal of Differential Equations (EJDE) [electronic only]

### A short note on ${L}^{q}$ theory for Stokes problem with a pressure-dependent viscosity

Czechoslovak Mathematical Journal

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$, namely, it is represented by a stress tensor $T\left(Du,p\right):=\nu \left(p,|D{|}^{2}\right)D$ which satisfies $r$-growth condition with $r\in \left(1,2\right]$. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in...

### A short note on regularity criteria for the Navier-Stokes equations containing the velocity gradient

Banach Center Publications

We review several regularity criteria for the Navier-Stokes equations and prove some new ones, containing different components of the velocity gradient.

### A stochastic lagrangian proof of global existence of the Navier-Stokes equations for flows with small Reynolds number

Annales de l'I.H.P. Analyse non linéaire

### A uniqueness result for a model for mixtures in the absence of external forces and interaction momentum

Applications of Mathematics

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ${\rho }_{i}$ of the fluids and their velocity fields ${u}^{\left(i\right)}$ are prescribed at infinity: ${\rho }_{i}{|}_{\infty }={\rho }_{i\infty }>0$, ${u}^{\left(i\right)}{|}_{\infty }=0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ${\rho }_{i}\equiv {\rho }_{i\infty }$, ${u}^{\left(i\right)}\equiv 0$, $i=1,2$.

### Additional note on partial regularity of weak solutions of the Navier-Stokes equations in the class ${L}^{\infty }\left(0,T,{L}^{3}{\left(\Omega \right)}^{3}\right)$

Applications of Mathematics

We present a simplified proof of a theorem proved recently concerning the number of singular points of weak solutions to the Navier-Stokes equations. If a weak solution $𝐮$ belongs to ${L}^{\infty }\left(0,T,{L}^{3}{\left(\Omega \right)}^{3}\right)$, then the set of all possible singular points of $𝐮$ in $\Omega$ is at most finite at every time ${t}_{0}\in \left(0,T\right)$.

### Almost global solutions of the free boundary problem for the equations of a magnetohydrodynamic incompressible fluid

Applicationes Mathematicae

Almost global in time existence of solutions for equations describing the motion of a magnetohydrodynamic incompressible fluid in a domain bounded by a free surfaced is proved. In the exterior domain we have an electromagnetic field which is generated by some currents which are located on a fixed boundary. We prove that a solution exists for t ∈ (0,T), where T > 0 is large if the data are small.

### An improved regularity criteria for the MHD system based on two components of the solution

Applications of Mathematics

As observed by Yamazaki, the third component ${b}_{3}$ of the magnetic field can be estimated by the corresponding component ${u}_{3}$ of the velocity field in ${L}^{\lambda }$