Partial regularity of solution to generalized Navier-Stokes problem

2014 ◽  
Vol 12 (10) ◽  
Author(s):  
Václav Mácha
Keyword(s):  
Partial Regularity ◽  
Stokes Problem ◽  
Regularity Of Solutions ◽  
Navier Stokes ◽  
Singular Set ◽  
Hölder Continuous ◽  
Indirect Approach ◽  
Regularity Of Solution ◽  
Dimension 2 ◽  
Deviatoric Part

AbstractIn 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.

scholarly journals On the regularity of solutions of one-dimensional variational obstacle problems

2018 ◽  
Vol 11 (2) ◽  
pp. 203-222 ◽  
Author(s):  
Jean-Philippe Mandallena
Keyword(s):  
Partial Regularity ◽  
Direct Method ◽  
Variational Problems ◽  
Obstacle Problems ◽  
Regularity Of Solutions ◽  
Hölder Continuous ◽  
One Dimensional ◽  
A Direct Method

AbstractWe study the regularity of solutions of one-dimensional variational obstacle problems in {W^{1,1}} when the Lagrangian is locally Hölder continuous and globally elliptic. In the spirit of the work of Sychev [5, 6, 7], a direct method is presented for investigating such regularity problems with obstacles. This consists of introducing a general subclass {\mathcal{L}} of {W^{1,1}}, related in a certain way to one-dimensional variational obstacle problems, such that every function of {\mathcal{L}} has Tonelli’s partial regularity, and then to prove that, depending on the regularity of the obstacles, solutions of corresponding variational problems belong to {\mathcal{L}}. As an application of this direct method, we prove that if the obstacles are {C^{1,\sigma}}, then every Sobolev solution has Tonelli’s partial regularity.

scholarly journals Partial regularity up to the boundary for solutions of subquadratic elliptic systems

2018 ◽  
Vol 7 (4) ◽  
pp. 469-483 ◽  
Author(s):  
Zhong Tan ◽  
Yanzhen Wang ◽  
Shuhong Chen
Keyword(s):  
Partial Regularity ◽  
Direct Method ◽  
Elliptic Systems ◽  
Divergence Form ◽  
Singular Set ◽  
Nonlinear Elliptic Systems ◽  
Hölder Continuous ◽  
Nonlinear Elliptic ◽  
Controllable Growth Condition ◽  
Controllable Growth

AbstractIn this paper, we are concerned with the nonlinear elliptic systems in divergence form under controllable growth condition. We prove that the weak solution u is locally Hölder continuous besides a singular set by using the direct method and classical Morrey-type estimates. Here the Hausdorff dimension of the singular set is less than {n-p}. This result not only holds in the interior, but also holds up to the boundary.

scholarly journals Partial regularity of stable solutions to the fractional Geľfand-Liouville equation

2021 ◽  
Vol 10 (1) ◽  
pp. 1316-1327
Author(s):  
Ali Hyder ◽  
Wen Yang
Keyword(s):  
Liouville Equation ◽  
Weak Solutions ◽  
Partial Regularity ◽  
Singular Set ◽  
Stable Solutions ◽  
Gelfand Problem

Abstract We analyze stable weak solutions to the fractional Geľfand problem ( − Δ ) s u = e u i n Ω ⊂ R n . $$\begin{array}{} \displaystyle (-{\it\Delta})^su = e^u\quad\mathrm{in}\quad {\it\Omega}\subset\mathbb{R}^n. \end{array}$$ We prove that the dimension of the singular set is at most n − 10s.

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