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A direct proof of the Caffarelli-Kohn-Nirenberg theorem

Jörg Wolf (2008)

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 l i m s u p R 0 1 / R Q R ( x , t ) | c u r l u × u / | u | | ² d x d t ε * for a sufficiently small ε * > 0 .

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

Jiří Neustupa (2014)

Mathematica Bohemica

We deal with a suitable weak solution ( 𝐯 , p ) to the Navier-Stokes equations in a domain Ω 3 . We refine the criterion for the local regularity of this solution at the point ( 𝐟 x 0 , t 0 ) , which uses the L 3 -norm of 𝐯 and the L 3 / 2 -norm of p in a shrinking backward parabolic neighbourhood of ( 𝐱 0 , t 0 ) . 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 ( 𝐱 0 , t 0 ) , respectively in a ”small” subset of this exterior, are considered. The consequence is that...

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

Petr Kučera, Zdeněk Skalák (2003)

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.

Jesús Ildefonso Díaz, Jean-Michel Rakotoson, Paul G. Schmidt (2007)


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 short note on L q theory for Stokes problem with a pressure-dependent viscosity

Václav Mácha (2016)

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 ( D u , p ) : = ν ( p , | D | 2 ) D which satisfies r -growth condition with r ( 1 , 2 ] . In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in...

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

Jens Frehse, Sonja Goj, Josef Málek (2005)

Applications of Mathematics

We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ρ i of the fluids and their velocity fields u ( i ) are prescribed at infinity: ρ i | = ρ i > 0 , u ( i ) | = 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 ρ i ρ i , u ( i ) 0 , i = 1 , 2 .

Additional note on partial regularity of weak solutions of the Navier-Stokes equations in the class L ( 0 , T , L 3 ( Ω ) 3 )

Zdeněk Skalák (2003)

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 ( 0 , T , L 3 ( Ω ) 3 ) , then the set of all possible singular points of 𝐮 in Ω is at most finite at every time t 0 ( 0 , T ) .

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

Piotr Kacprzyk (2004)

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.

Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling ν ( p , · ) + as p +

M. Bulíček, Josef Málek, Kumbakonam R. Rajagopal (2009)

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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