203 research outputs found

### 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systems.Comment: This paper has been withdrawn by the author due to a crucial erro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 $\mu$HS 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 $L^2$ 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

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