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1
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Partial regularity of solution to generalized Navier-Stokes problem

100%
Mácha V.
Open Mathematics
|
2014
|
tom 12
|
nr 10
1460-1483
EN
In the presented work, we study the regularity of solutions to the generalized Navier-Stokes problem up to a C 2 boundary in dimensions two and three. The point of our generalization is an assumption that a deviatoric part of a stress tensor depends on a shear rate and on a pressure. We focus on estimates of the Hausdorff measure of a singular set which is defined as a complement of a set where a solution is Hölder continuous. We use so-called indirect approach to show partial regularity, for dimension 2 we get even an empty set of singular points.
2
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Boundary regularity of flows under perfect slip boundary conditions

84%
Kaplický P., Tichý J.
Open Mathematics
|
2013
|
tom 11
|
nr 7
1243-1263
EN
We investigate boundary regularity of solutions of generalized Stokes equations. The problem is complemented with perfect slip boundary conditions and we assume that the nonlinear elliptic operator satisfies non-standard ϕ-growth conditions. We show the existence of second derivatives of velocity and their optimal regularity.
3
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On a generalized Stokes problem

84%
Mácha V.
Open Mathematics
|
2011
|
tom 9
|
nr 4
874-887
EN
We deal with a generalization of the Stokes system. Instead of the Laplace operator, we consider a general elliptic operator and a pressure gradient with small perturbations. We investigate the existence and uniqueness of a solution as well its regularity properties. Two types of regularity are provided. Aside from the classical Hilbert regularity, we also prove the Hölder regularity for coefficients in VMO space.
4
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Boundary value problems for quasi-linear elliptic second order equations in unbounded cone-like domains

84%
Borsuk M., Wiśniewski D.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2051-2072
EN
We study the behaviour of weak solutions (as well as their gradients) of boundary value problems for quasi-linear elliptic divergence equations in domains extending to infinity along a cone.
5
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Conformal Geometry and the Composite Membrane Problem

84%
Chanillo S.
Analysis and Geometry in Metric Spaces
|
2013
|
tom 1
31-35
EN
We show that a certain eigenvalue minimization problem in two dimensions for the Laplace operator in conformal classes is equivalent to the composite membrane problem. We again establish such a link in higher dimensions for eigenvalue problems stemming from the critical GJMS operators. New free boundary problems of unstable type arise in higher dimensions linked to the critical GJMS operator. In dimension four, the critical GJMS operator is exactly the Paneitz operator.
6
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Global and exponential attractors for a Caginalp type phase-field problem

69%
Bangola B.
Open Mathematics
|
2013
|
tom 11
|
nr 9
1651-1676
EN
We deal with a generalization of the Caginalp phase-field model associated with Neumann boundary conditions. We prove that the problem is well posed, before studying the long time behavior of solutions. We establish the existence of the global attractor, but also of exponential attractors. Finally, we study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist.
7
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Sobolev-Kantorovich Inequalities

69%
Ledoux M.
Analysis and Geometry in Metric Spaces
|
2015
|
tom 3
|
nr 1
EN
In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.
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