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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 ) .

A continuity property for the inverse of Mañé's projection

Zdeněk Skalák — 1998

Applications of Mathematics

Let X be a compact subset of a separable Hilbert space H with finite fractal dimension d F ( X ) , and P 0 an orthogonal projection in H of rank greater than or equal to 2 d F ( X ) + 1 . For every δ > 0 , there exists an orthogonal projection P in H of the same rank as P 0 , which is injective when restricted to X and such that P - P 0 < δ . This result follows from Mañé’s paper. Thus the inverse ( P | X ) - 1 of the restricted mapping P | X X P X is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé’s...

Conditions of Prodi-Serrin's type for local regularity of suitable weak solutions to the Navier-Stokes equations

Zdeněk Skalák — 2002

Commentationes Mathematicae Universitatis Carolinae

In the context of suitable weak solutions to the Navier-Stokes equations we present local conditions of Prodi-Serrin’s type on velocity 𝐯 and pressure p under which ( 𝐱 0 , t 0 ) Ω × ( 0 , T ) is a regular point of 𝐯 . The conditions are imposed exclusively on the outside of a sufficiently narrow space-time paraboloid with the vertex ( 𝐱 0 , t 0 ) and the axis parallel with the t -axis.

An existence theorem for the Boussinesq equations with non-Dirichlet boundary conditions

Zdeněk SkalákPetr Kučera — 2000

Applications of Mathematics

The evolution Boussinesq equations describe the evolution of the temperature and velocity fields of viscous incompressible Newtonian fluids. Very often, they are a reasonable model to render relevant phenomena of flows in which the thermal effects play an essential role. In the paper we prescribe non-Dirichlet boundary conditions on a part of the boundary and prove the existence and uniqueness of solutions to the Boussinesq equations on a (short) time interval. The length of the time interval depends...

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