Global Well-Posedness and Analyticity of the Primitive Equations of Geophysics in Variable Exponent Fourier–Besov Spaces

We consider the Cauchy problem of the three-dimensional primitive equations of geophysics. By using the Littlewood–Paley decomposition theory and Fourier localization technique, we prove the global well-posedness for the Cauchy problem with the Prandtl number P=1 in variable exponent Fourier–Besov spaces for small initial data in these spaces. In addition, we prove the Gevrey class regularity of the solution. For the primitive equations of geophysics, our results can be considered as a symmetry in variable exponent Fourier–Besov spaces.

Well-posedness, wave breaking, Holder continuity and periodic peakons for a nonlocal sine-µ-Camassa-Holm equation

In this paper, we investigate the initial value problem of a nonlocal sine-type µ-Camassa-Holm (µCH) equation, which is the µ-version of the sine-type CH equation. We first discuss its local well-posedness in the framework of Besov spaces. Then a sufficient condition on the initial data is provided to ensure the occurance of the wave-breaking phenomenon. We finally prove the H¨older continuity of the data-to-solution map, and find the explicit formula of the global weak periodic peakon solution.

Norms of inclusions between some spaces of analytic functions

The inclusions between the Besov spaces $$B^q$$ B q , the Bloch space $$\mathcal {B}$$ B and the standard weighted Bergman spaces $$A^p_{\alpha}$$ A α p are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.

The Well Posedness for Nonhomogeneous Boussinesq Equations

This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces (θ,u)∈LT∞(B˙p,1N/p)×LT∞(B˙p,1N/p−1)⋂LT1(B˙p,1N/p+1) with 1<p<2N. We proved the global existence of the solution when the initial velocity is small with respect to the viscosity, as well as the initial temperature approaches a positive constant. Furthermore, we proved the uniqueness for 1<p≤N. Our results can been seen as a version of symmetry in Besov space for the Boussinesq equations.

Multivariate wavelet leaders Rényi dimension and multifractal formalism in mixed Besov spaces

In this paper, we first establish a general lower bound for the multivariate wavelet leaders Rényi dimension valid for any pair [Formula: see text] of functions on [Formula: see text] where [Formula: see text] belongs to the Besov space [Formula: see text] with [Formula: see text] and [Formula: see text] belongs to [Formula: see text] with [Formula: see text]. We then prove the optimality of this result for quasi all pairs [Formula: see text] in the Baire generic sense. Finally, we compute both iso-mixed and upper-multivariate Hölder spectra for all pairs [Formula: see text] in the same [Formula: see text]-set. This allows to prove (respectively, study) the Baire generic validity of the upper-multivariate (respectively, iso-multivariate) multifractal formalism based on wavelet leaders for such pairs.