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The present paper is the continuation of work [18] , devoted to the study of an inviscid zero-Mach number system in the framework of endpoint Besov spaces of type B ∞ , r s ( R d ) , r ∈ [ 1 , ∞ ] , d ≥ 2 , which can be embedded in the Lipschitz class C 0 , 1 . In particular, the largest case B ∞ , 1 1 and the case of Hölder spaces C 1 , α are taken into account. The local in time well-posedness of this system is proved, under an additional finite-energy hypothesis on the initial data. The key to get this result is new a priori estimates for parabolic equations with variable coefficients in endpoint spaces B ∞ , r s ( R d ) , which are of independent interest. In the special case of space dimension d = 2 , we are able to give a lower bound for the lifespan, such that the solutions tend to be globally defined when the initial inhomogeneity is small. There, we will show refined a priori estimates in endpoint Besov spaces for transport equations. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of a three-component Camassa–Holm system. We first prove the local well-posedness of the system in Besov spaces B p , r s with p , r ∈ [ 1 , ∞ ] , s > max ⁡ { 1 p , 1 2 } by using the Littlewood–Paley theory and transport equations theory. Then, we establish two blow-up criteria which along with the conservation laws enable us to study global existence. Moreover, if the initial data satisfies some certain sign conditions, we obtain a global existence result. Finally, we verify that the system possesses peakon solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the global well-posedness of the Cauchy problem of the 3D incompressible nematic liquid crystal system with initial data in the critical Besov space B2,11/2 (R³) × B2,13/2 (R³). In particular, we prove that there exists a positive constant C0 such that the nematic liquid crystal system has a unique global solution with initial data (u0, d0) = (u0h, u0³, d0) which satisfies (...} ≤ c0, for some c0 sufficiently small, where d0 is a constant vector with |d0| = 1. Here ν and μ are two positive viscosity constants. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corresponding result in [4], the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estimate in terms of norm is essential for this result. Contrary to the usual MHD the global well-posedness in the dimensional Hall-MHD is wide open. [Copyright &y& Elsevier]

Abstract: In this paper we consider the Cauchy problem for the generalized Camassa–Holm equation in Besov space. First, we prove that the solutions to the Cauchy problem for the generalized Camassa–Holm equation do not depend uniformly continuously on the initial data in with when . Second, combining the real interpolations among inhomogeneous Besov spaces with Lemma 5.2.1 of [6] which is called Osgood Lemma (a substitute for Gronwall inequality), we show that the Cauchy problem for the generalized Camassa–Holm equation is locally well-posed in . Finally, we give a blow-up criterion. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces , , (which generalize the Sobolev spaces ) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with , , ) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated. [Copyright &y& Elsevier]

Abstract: Considered in this paper is the modified Camassa–Holm equation with cubic nonlinearity, which is integrable and admits the single peaked solitons and multi-peakon solutions. The short-wave limit of this equation is known as the short-pulse equation. The main investigation is the Cauchy problem of the modified Camassa–Holm equation with qualitative properties of its solutions. It is firstly shown that the equation is locally well-posed in a range of the Besov spaces. The blow-up scenario and the lower bound of the maximal time of existence are then determined. A blow-up mechanism for solutions with certain initial profiles is described in detail and nonexistence of the smooth traveling wave solutions is also demonstrated. [Copyright &y& Elsevier]

Abstract: The well-posedness of generalized Navier–Stokes equations with initial data in some critical homogeneous Besov spaces and in some critical Q spaces was known. In this paper, we establish a wavelet characterization of Besov type Morrey spaces under the action of semigroup. As an application, we obtain the well-posedness of smooth solution for the generalized Navier–Stokes equations with initial data in some critical homogeneous Besov type Morrey spaces (, ), and or , and , with divergence free. These critical homogeneous Besov type Morrey spaces are larger than corresponding classical Besov spaces and cover Q spaces. [Copyright &y& Elsevier]

Abstract: In this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via “caloric extension”. We apply our result to the Navier–Stokes equations, the surface quasi-geostrophic equations, the Kuramoto–Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier–Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation. [Copyright &y& Elsevier]

Abstract: In this paper we consider the Cauchy problem for the integrable Novikov equation. By using the Littlewood–Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the integrable Novikov equation is locally well-posed in the Besov space with and . In particular, when with and , for all , we have that . We also prove that the local well-posedness of the Cauchy problem for the Novikov equation fails in . [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem for a two-component Degasperis–Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented. [Copyright &y& Elsevier]

Abstract: In this paper, we study a multidimensional bipolar hydrodynamic model for semiconductors or plasmas. This system takes the form of the bipolar Euler–Poisson model with electric field and frictional damping added to the momentum equations. In the framework of the Besov space theory, we establish the global existence of smooth solutions for Cauchy problems when the initial data are sufficiently close to the constant equilibrium. Next, based on the special structure of the nonlinear system, we also show the uniform estimate of solutions with respect to the relaxation time by the high- and low-frequency decomposition methods. Finally we discuss the relaxation-time limit by compact arguments. That is, it is shown that the scaled classical solution strongly converges towards that of the corresponding bipolar drift-diffusion model, as the relaxation time tends to zero. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain classical solutions of a time-dependent Ginzburg–Landau (TDGL) equations come from the superfluid atomic Fermi gases near the Feshbach resonance from the fermion–boson model. By the properties of Besov and Sobolev spaces and matrix theory, together with the energy method, we establish the global existence and uniqueness of classical solutions of the TDGL equations in the case of BCS-BEC crossover. [Copyright &y& Elsevier]

Abstract: Recently, Novikov found a new integrable equation (we call it the Novikov equation in this paper), which has nonlinear terms that are cubic, rather than quadratic, and admits peaked soliton solutions (peakons). Firstly, we prove that the Cauchy problem for the Novikov equation is locally well-posed in the Besov spaces (which generalize the Sobolev spaces ) with the critical index . Then, well-posedness in with , is also established by applying Kato''s semigroup theory. Finally, we present two results on the persistence properties of the strong solution for the Novikov equation. [Copyright &y& Elsevier]

Abstract: In this paper we study a fractional diffusion Boussinesq model which couples a Navier–Stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature. We establish global well-posedness results with rough initial data. [Copyright &y& Elsevier]

Abstract: The present paper is devoted to the study of the well-posedness issue for the density-dependent Euler equations in the whole space. We establish local-in-time results for the Cauchy problem pertaining to data in the Besov spaces embedded in the set of Lipschitz functions, including the borderline case . A continuation criterion in the spirit of the celebrated one by Beale, Kato and Majda (1984) in for the classical Euler equations, is also proved. In contrast with the previous work dedicated to this system in the whole space, our approach is not restricted to the framework or to small perturbations of a constant density state: we just need the density to be bounded away from zero. The key to that improvement is a new a priori estimate in Besov spaces for an elliptic equation with nonconstant coefficients. [Copyright &y& Elsevier]

Abstract: The compressible Navier–Stokes–Poisson system is concerned in the present paper, and the global existence and uniqueness of the strong solution is shown in the framework of hybrid Besov spaces in three and higher dimensions. [Copyright &y& Elsevier]

Abstract: In this paper we study the model of heat transfer in a porous medium with a critical diffusion. We obtain global existence and uniqueness of solutions to the equations of heat transfer of incompressible fluid in Besov spaces with by the method of modulus of continuity and Fourier localization technique. [Copyright &y& Elsevier]