DCDS
Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions
Wei Luo Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2016, 36(9): 5047-5066 doi: 10.3934/dcds.2016019
In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using Littlewood-Paley theory and transport equations theory, we establish the local well-posedness of the system in the critical Besov space. Moreover, we obtain some weighted $L^p$ estimates of strong solutions to the system. By taking suitable weighted functions, we can get the persistence properties of strong solutions on exponential, algebraic and logarithmic decay rates, respectively.
keywords: critical Besov space local well-posedness persistence properties. A three-component Camassa-Holm system
DCDS
On the initial value problem for higher dimensional Camassa-Holm equations
Kai Yan Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2015, 35(3): 1327-1358 doi: 10.3934/dcds.2015.35.1327
This paper is concerned with the the initial value problem for higher dimensional Camassa-Holm equations. Firstly, the local well-posedness for this equations in both supercritical and critical Besov spaces are established. Then two blow-up criterions of strong solutions to the equations are derived. Finally, the analyticity of its solutions is proved in both variables, globally in space and locally in time.
keywords: analytic solutions. blow up Higher dimensional Camassa-Holm equations local well-posedness Besov spaces initial value problem
CPAA
Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation
Zhaoyang Yin
Communications on Pure & Applied Analysis 2004, 3(3): 501-508 doi: 10.3934/cpaa.2004.3.501
We establish the local well-posedness for the periodic generalized Camassa-Holm equation. We also give the precise blow-up scenario and prove that the equation has smooth solutions that blow up in finite time.
keywords: The periodic generalized Camassa-Holm equation blow-up scenario. local well- posedness
DCDS
Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation
Min Li Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2017, 37(12): 6471-6485 doi: 10.3934/dcds.2017280

In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in $H^s (\mathbb{S}), s > \frac{3}{2},$ by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.

keywords: An integrable dispersive Hunter-Saxton equation the Kato method Blow-up travelling wave solutions the sinh-Gordon equation
DCDS
Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data
Xiaoping Zhai Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2017, 37(5): 2829-2859 doi: 10.3934/dcds.2017122

In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in $\mathbb{R}^d \, (d≥2)$ by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on $\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$, $\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$ and $\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$. This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in $\mathbb{R}^2$ provided the initial cell density $n_0$ and the initial chemical concentration $c_0$ are doubly exponential small compared with the initial velocity field $u_0$.

keywords: The Chemotaxis-Navier-Stokes equations global wellposedness Besov spaces
DCDS
On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators
Huijun He Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1509-1537 doi: 10.3934/dcds.2017062

In this paper, we mainly consider the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators: $m=(1-\partial_x^2)^su, s>1$. By Littlewood-Paley theory and transport equation theory, we first establish the local well-posedness of the generalized b-equation with fractional higher-order inertia operators which is the subsystem of the generalized two-component water wave system. Then we prove the local well-posedness of the generalized two-component water wave system with fractional higher-order inertia operators. Next, we present the blow-up criteria for these systems. Moreover, we obtain some global existence results for these systems.

keywords: Two-component shallow water wave system with fractional higher-order inertia operators Littlewood-Paley theory local well-posedness blow-up criteria global existence
DCDS
On the Cauchy problem for a four-component Camassa-Holm type system
Zeng Zhang Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2015, 35(10): 5153-5169 doi: 10.3934/dcds.2015.35.5153
This paper is concerned with a four-component Camassa-Holm type system proposed in [37], where its bi-Hamiltonian structure and infinitely many conserved quantities were constructed. In the paper, we first establish the local well-posedness for the system. Then we present several global existence and blow-up results for two integrable two-component subsystems.
keywords: A four-component Camassa-Holm system local well-posedness global existence blow-up.
DCDS
Well-posedness, blowup, and global existence for an integrable shallow water equation
Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2004, 11(2&3): 393-411 doi: 10.3934/dcds.2004.11.393
We establish the local well-posedness for a recently derived model that combines the linear dispersion of Korteweg-de Veris equation with the nonlinear/nonlocal dispersion of the Camassa-Holm equation, and we prove that the equation has solutions that exist for indefinite times as well as solutions that blow up in finite time. We also derive an explosion criterion for the equation, and we give a sharp estimate of the existence time for solutions with smooth initial data.
keywords: lower semicontinuity peaked solitons. explosion criterion Local well-posedness sharp estimate from below blowup global existence
DCDS
The Cauchy problem for a generalized Novikov equation
Rudong Zheng Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 3503-3519 doi: 10.3934/dcds.2017149

We establish the local well-posedness for a generalized Novikov equation in nonhomogeneous Besov spaces. Besides, we obtain a blow-up criteria and provide a sufficient condition for strong solutions to blow up in finite time.

keywords: A generalized Novikov equation local well-posedness Bseov spaces blow up
DCDS
Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions
Xi Tu Zhaoyang Yin
Discrete & Continuous Dynamical Systems - A 2016, 36(5): 2781-2801 doi: 10.3934/dcds.2016.36.2781
In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the equation in Besov spaces. Then we give a blow-up criterion for the Cauchy problem of the equation. we present a blow-up result and the exact blow-up rate of strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion. Finally, we verify that the system possesses peakon solutions.
keywords: blow-up A generalized Camassa-Holm equation Besov spaces local well-posedness peakon solutions.

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