CPAA
Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation
Zhaoyang Yin
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
On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators
Huijun He Zhaoyang Yin

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
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
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
Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions
Xi Tu Zhaoyang Yin
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.
DCDS
Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation
Jinlu Li Zhaoyang Yin
We first establish the local existence and uniqueness of strong solutions for the Cauchy problem of a generalized Camassa-Holm equation in nonhomogeneous Besov spaces by using the Littlewood-Paley theory. Then, we prove that the solution depends continuously on the initial data in the corresponding Besov space. Finally, we derive a blow-up criterion and present a blow-up result and a blow-up rate of the blow-up solutions to the equation.
keywords: A generalized Camassa-Holm equation blow-up rate. blow-up local well-posedness
DCDS
Well-posedness for a modified two-component Camassa-Holm system in critical spaces
Kai Yan Zhaoyang Yin
This paper is concerned with the problem of well-posedness for a modified two-component Camassa-Holm system in Besov spaces with the critical index $s=\frac 3 2$.
keywords: Modified two-component Camassa-Holm system Besov spaces critical index. well-posedness
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
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
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
DCDS
Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data
Xiaoping Zhai Zhaoyang Yin

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

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