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}

Exponential Change of Measure for General Piecewise Deterministic Markov Processes

We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant 0}$ with measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continuous. A general form of the exponential martingales is presented as $$M^f_t=\frac{f(X_t)}{f(X_0)}\left[\mathrm{Sexp}\left(\int_{(0,t]}\frac{\mathrm{d}L(\mathcal{A}f)_s}{f(X_{s-})}\right)\right]^{-1}.$$ Using this exponential martingale as a likelihood ratio process, we define a new probability measure. It is shown that the original process remains a general PDMP under the new probability measure. And we find the new measure-valued generator and its domain

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prove several new global existence results for these two-component shallow water systems.Comment: This paper has been withdrawn by the author due to a crucial erro

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

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

On the Cauchy problem of a periodic 2-component μ\mu-Hunter-Saxton equation

In this paper, we study the Cauchy problem of a periodic 2-component $\mu$-Hunter-Saxton system. We first establish the local well-posedness for the periodic 2-component $\mu$-Hunter-Saxton system by Kato's semigroup theory. Then, we derive precise blow-up scenarios for strong solutions to the system. Moreover, we present a blow-up result for strong solutions to the system. Finally, we give a global existence result to the system

Global existence and well-posedness of 2D viscous shallow water system in Sobolev spaces with low regularity

In this paper we consider the Cauchy problem for 2D viscous shallow water system in $H^s(\mathbb{R}^2)$, $s>1$. We first prove the local well-posedness of this problem by using the Littlewood-Paley theory, the Bony decomposition, and the theories of transport equations and transport diffusion equations. Then, we get the global existence of the system with small initial data in $H^s(\mathbb{R}^2)$, $s>1$. Our obtained result improves the recent result in \cite{W}Comment: arXiv admin note: substantial text overlap with arXiv:1402.492

On the Cauchy problem of a weakly dissipative μ\muHS equation

In this paper, we study the Cauchy problem of a weakly dissipative $\mu$HS equation. We first establish the local well-posedness for the weakly dissipative $\mu$HS equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation

Global weak solutions for a periodic two-component μ\mu-Hunter-Saxton system

This paper is concerned with global existence of weak solution for a periodic two-component $\mu$-Hunter-Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component $\mu$-Hunter-Saxton system