Well-posedness and global existence of 2D viscous shallow water system in Besov spaces

In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(\mathbb{R}^2)$, $s>max\{1,\frac{2}{p}\}$, $1\leq p,r\leq \infty$ by using the Littlewood-Paley theory, the Bony decomposition and the theories of transport equations and transport diffusion equations. Then we can prove the global existence of the system with small enough initial data in $B^s_{p,r}(\mathbb{R}^2)$, $1\leq p\leq2$, $1\leq r<\infty$ and $s>\frac{2}{p}$. Our obtained results generalize and cover the recent results in \cite{W}

Global existence for the two-component Camassa-Holm system and the modified two-component Camassa-Holm system

The present work is mainly concerned with global existence for the two-component Camassa-Holm system and the modified two-component Camassa-Holm system. By discovering new conservative quantities of these systems, we prove several new global existence results for these two-component shallow water systems.Comment: This paper has been withdrawn by the author due to a crucial erro

Global weak solutions to a weakly dissipative μ\muHS equation

This paper is concerned with global existence of weak solutions for a weakly dissipative $\mu$HS equation by using smooth approximate to initial data and Helly$^{,}$s theorem

Global Well-posedness for the Generalized Navier-Stokes System

In this paper we investigate well-posedness of the Cauchy problem of the three dimensional generalized Navier-Stokes system. We first establish local well-posedness of the GNS system for any initial data in the Fourier-Herz space $\chi^{-1}$. Then we show that if the $\chi^{-1}$ norm of the initial data is smaller than C$\nu$ in the GNS system where $\nu$ is the viscosity coefficient, the corresponding solution exists globally in time. Moreover, we prove global well-posedness of the Navier-Stokes system without norm restrictions on the corresponding solutions provided the $\chi^{-1}$ norm of the initial data is less than $\nu.$ Our obtained results cover and improve recent results in \cite{Zhen Lei,wu}

On the Cauchy problem of a two-component b-family equation

In this paper, we study the Cauchy problem of a two-component b-family equation. We first establish the local well-posedness for a two-component b-family equation by Kato's semigroup theory. Then, we derive precise blow-up scenarios for strong solutions to the equation. Moreover, we present several blow-up results for strong solutions to the equation

Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations

This paper is concerned with a class of nonlocal dispersive models -- the $\theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: $$(1-\partial_x^2)u_t+(1-\theta\partial_x^2)(\frac{u^2}{2})_x =(1-4\theta)(\frac{u_x^2}{2})_x,$$ including integrable equations such as the Camassa-Holm equation, $\theta=1/3$, and the Degasperis-Procesi equation, $\theta=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $\theta \in \mathbb{R}$. It is shown that as $\theta$ increases regularity of solutions improves: (i) $0 <\theta < 1/4$, the solution will blow up when the momentum of initial data satisfies certain sign conditions; (ii) $1/4 \leq \theta < 1/2$, the solution will blow up when the slope of initial data is negative at one point; (iii) ${1/2} \leq \theta \leq 1$ and $\theta=\frac{2n}{2n-1}, n\in \mathbb{N}$, global existence of strong solutions is ensured. Moreover, if the momentum of initial data has a definite sign, then for any $\theta\in \mathbb{R}$ global smoothness of the corresponding solution is proved. Proofs are either based on the use of some global invariants or based on exploration of favorable sign conditions of quantities involving solution derivatives. Existence and uniqueness results of global weak solutions for any $\theta \in \mathbb{R}$ are also presented. For some restricted range of parameters results here are equivalent to those known for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, {\it J. reine angew. Math.}, {\bf 624} (2008)51--80.]Comment: 21 page

Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space

In this paper we mainly investigate the Cauchy problem of a two-component Novikov system. We first prove the local well-posedness of the system in Besov spaces $B^{s-1}_{p,r}\times B^s_{p,r}$ with $p,r\in[1,\infty],~s>\max\{1+\frac{1}{p},\frac{3}{2}\}$ by using the Littlewood-Paley theory and transport equations theory. Then, by virtue of logarithmic interpolation inequalities and the Osgood lemma, we establish the local well-posedness of the system in the critical Besov space $B^{\frac{1}{2}}_{2,1}\times B^{\frac{3}{2}}_{2,1}$. Moreover, we present two blow-up criteria for the system by making use of the conservation laws.Comment: arXiv admin note: text overlap with arXiv:1505.0008

The L2L^2 decay for the 2D co-rotation FENE dumbbell model of polymeric flows

In this paper we mainly study the long time behaviour of solutions to the finite extensible nonlinear elastic (FENE) dumbbell model with dimension two in the co-rotation case. Firstly, we obtain the $L^2$ decay rate of the velocity of the 2D co-rotation FENE model is $(1+t)^{-\frac{1}{2}}$ with small data. Then, by virtue of the Littlewood-Paley theory, we can remove the small condition. Our obtained sharp result improves considerably the recent results in \cite{Luo-Yin,Schonbek}.Comment: arXiv admin note: text overlap with arXiv:1603.0414

Gevrey regularity and analyticity for Camassa-Holm type systems

In this paper we mainly investigate the Cauchy problem of some Camassa-Holm type systems. By constructing a new auxiliary function, we present a generalized Ovsyannikov theorem. By using this theorem and the basic properties of Sobolev-Gevrey spaces, we prove the Gevrey regularity and analyticity of these systems. Moreover, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.Comment: arXiv admin note: text overlap with arXiv:1506.0118

Global Weak Solution for a generalized Camassa-Holm equation

In this paper we mainly investigate the Cauchy problem of a generalized Camassa-Holm equation. First by this relationship between the Degasperis-Procesi equation and the generalized Camassa-Holm equation, we then obtain two global existences result and two blow-up result. Then, we prove the existence and uniqueness of global weak solutions.Comment: arXiv admin note: text overlap with arXiv:1505.00086, arXiv:1511.0231