scholarly journals Uniqueness of Weak Solutions to an Electrohydrodynamics Model

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yong Zhou ◽  
Jishan Fan
Keyword(s):  
Besov Space ◽  
Weak Solutions ◽  
Uniqueness Result ◽  
Strong Uniqueness ◽  
Homogeneous Besov Space

This paper studies uniqueness of weak solutions to an electrohydrodynamics model inℝd(d=2,3). Whend=2, we prove a uniqueness without any condition on the velocity. Ford=3, we prove a weak-strong uniqueness result with a condition on the vorticity in the homogeneous Besov space.

scholarly journals Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Zhaohui Dai ◽  
Xiaosong Wang ◽  
Lingrui Zhang ◽  
Wei Hou
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Interpolation Inequality ◽  
Embedding Theorem ◽  
The Earth ◽  
The Cauchy Problem ◽  
Homogeneous Besov Space ◽  
Blow Up Criterion

The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressurepin the homogeneous Besov spaceḂ∞,∞0.

scholarly journals SOBOLEV INEQUALITIES WITH SYMMETRY

2009 ◽  
Vol 11 (03) ◽  
pp. 355-365 ◽  
Author(s):  
YONGGEUN CHO ◽  
TOHRU OZAWA
Keyword(s):  
Besov Space ◽  
Symmetric Functions ◽  
Sobolev Inequalities ◽  
End Point ◽  
Compact Embeddings ◽  
Homogeneous Besov Space

In this paper, we derive some Sobolev inequalities for radially symmetric functions in Ḣs with 1/2 < s < n/2. We show the end point case s = 1/2 on the homogeneous Besov space [Formula: see text]. These results are extensions of the well-known Strauss' inequality [13]. Also non-radial extensions are given, which show that compact embeddings are possible in some Lp spaces if a suitable angular regularity is imposed.

scholarly journals Uniqueness of weak solution for nonlinear elliptic equations in divergence form

2000 ◽  
Vol 23 (5) ◽  
pp. 313-318 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Uniqueness Result ◽  
Quasilinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Quasilinear Elliptic

We study the uniqueness of weak solutions for quasilinear elliptic equations in divergence form. Some counterexamples are given to show that our uniqueness result cannot be improved in the general case.

scholarly journals Logarithmically Improved Regularity Criterion for the 3D Micropolar Fluid Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Hui Zhang
Keyword(s):  
Besov Space ◽  
Strong Solution ◽  
Weak Solutions ◽  
Micropolar Fluid ◽  
Regularity Criterion ◽  
Vorticity Field ◽  
Regularity Of Weak Solutions ◽  
Fluid Equations ◽  
Incompressible Micropolar Fluid ◽  
Micropolar Fluid Equations

We study the regularity of weak solutions to the incompressible micropolar fluid equations. We obtain an improved regularity criterion in terms of vorticity of velocity in Besov space. It is proved that if the vorticity field satisfies ∫0T∇×uB˙∞,∞0/1+log1+∇×uB˙∞,∞0dt<∞ then the strong solution can be smoothly extended after time T.

Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian

2017 ◽  
Vol 17 (4) ◽  
pp. 837-839 ◽  
Author(s):  
Umberto Biccari ◽  
Mahamadi Warma ◽  
Enrique Zuazua
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Space ◽  
Besov Space ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Local Regularity ◽  
Elliptic Regularity ◽  
Open Set ◽  
Regularity Of Weak Solutions

AbstractIn [1], for {1<p<\infty}, we proved the {W^{2s,p}_{\mathrm{loc}}} local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian {(-\Delta)^{s}} on an arbitrary bounded open set of {\mathbb{R}^{N}}. Here we make a more precise and rigorous statement. In fact, for {1<p<2} and {s\neq\frac{1}{2}}, local regularity does not hold in the Sobolev space {W^{2s,p}_{\mathrm{loc}}}, but rather in the larger Besov space {(B^{2s}_{p,2})_{\mathrm{loc}}}.

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