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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 optimal control problem for a generalized Boussinesq model: The time dependent case.

Revista Matemática Complutense

### Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu \left(p,·\right)\to +\infty$ as $p\to +\infty$

Czechoslovak Mathematical Journal

Over a large range of the pressure, one cannot ignore the fact that the viscosity grows significantly (even exponentially) with increasing pressure. This paper concerns long-time and large-data existence results for a generalization of the Navier-Stokes fluid whose viscosity depends on the shear rate and the pressure. The novelty of this result stems from the fact that we allow the viscosity to be an unbounded function of pressure as it becomes infinite. In order to include a large class of viscosities...

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