Optimality of Serrin type extension criteria to the Navier-Stokes equations

We prove that a strong solution u to the Navier-Stokes equations on (0, T) can be extended if either u ∈ L θ (0, T; U ˙ ∞ , 1 / θ , ∞ − α $\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$ ) for 2/θ + α = 1, 0 < α < 1 or u ∈ L 2(0, T; V ˙ ∞ , ∞ , 2 0 $\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$ ), where U ˙ p , β , σ s $\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$ and V ˙ p , q , θ s $\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$ are Banach spaces that may be larger than the homogeneous Besov space B ˙ p , q s $\begin{array}{} \displaystyle \dot{B}^{s}_{p,q} \end{array}$ . Our method is based on a bilinear estimate and a logarithmic interpolation inequality.

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

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.

A Regularity Criterion for the Magneto-Micropolar Fluid Equations inB˙∞,∞−1

The paper is dedicated to study of the Cauchy problem for the magneto-micropolar fluid equations in three-dimensional spaces. A new logarithmically improved regularity criterion for the magneto-micropolar fluid equations is established in terms of the pressure in the homogeneous Besov spaceB˙∞,∞−1.

Wavelet Shrinkage Based Variational Image Decomposition Model

A new class of variational models based on Besov spaces B1,1s(s>0 ) and homogeneous Besov space E=B∞,∞-1for image decomposition is proposed. The proposed models can be regarded as generalizations of Aujol-Chambolle model. The associated minimizers of variational problems can be expressed by applying different shrinkage functions which depend on the wavelet scale to each wavelet coefficient. The wavelet based treatment simplifies computation of this class of variational models. Finally, we present numerical results on denoising of both real and remote sensing images.

Uniqueness of Weak Solutions to an Electrohydrodynamics Model

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.

Homogeneous Besov Spaces on Stratified Lie Groups and Their Wavelet Characterization

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov spaceB˙p,qsin terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spacesB˙p,qswith1≤p,q<∞ands∈ℝ.

SOBOLEV INEQUALITIES WITH SYMMETRY

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.