Showing 1–20 of 111 results
Kai Yan, Zhaoyang Yin
This paper is concerned with the Cauchy problem for a two-component Degasperis-Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented.
Teng-Fei Zhang, Zhaoyang Yin
In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for $C^\infty$ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto's recent paper (See Morimoto: J. Pseudo-Differ. Oper. Appl. (2010) 1: 139-159, DOI:10.1007/s11868-010-0008-z), but we extend the range of the index $γ$ satisfying $γ+ 2s \in (-1,1)$, $s\in (0,1/2)$ and in this case we consider the kinetic factor in the form of $Φ(v)=|v|^γ$ instead of $\la v \ra ^γ$ as Morimoto did before.
Wei Luo, Zhaoyang Yin
In this paper we mainly investigate the Cauchy problem of the finite extensible nonlinear elastic (FENE) dumbbell model with dimension $d\geq2$. We first proved the local well-posedness for the FENE model in Besov spaces by using the Littlewood-Paley theory. Then by an accurate estimate we get a blow-up criterion. Moreover, if the initial data is perturbation around equilibrium, we obtain a global existence result. Our obtained results generalize recent results in [8].
Xingxing Liu, Zhaoyang Yin
In this paper, we investigate the orbital stability of peakons for a modified Camassa-Holm equation with cubic nonlinearity derived from the two-dimensional Euler equation. By overcoming the difficulties caused by one of the complicated conservation laws and introducing a new auxiliary function, we prove the orbital stability of peakons for the equation.
Zeng Zhang, Zhaoyang Yin
In this paper, we mainly study the Cauchy problem of the Euler-Nernst-Planck-Possion ($ENPP$) system. We first establish local well-posedness for the Cauchy problem of the $ENPP$ system in Besov spaces. Then we present a blow-up criterion of solutions to the $ENPP$ system. Moreover, we prove that the solutions of the Navier-Stokes-Nernst-Planck-Possion system converge to the solutions of the $ENPP$ system as the viscosity $ν$ goes to zero, and that the convergence rate is at least of order $ν^\frac{1}{2}$.
Zeng Zhang, Zhaoyang Yin
In this paper, we study the Cauchy problem of the Euler-Nernst-Planck-Possion system. We obtain global well-posedness for the system in dimension $d=2$ for any initial data in $H^{s_1}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2)\times H^{s_2}(\mathbb{R}^2)$ under certain conditions of $s_1$ and $s_2$.
Wenke Tan, Zhaoyang Yin
In this paper, we mainly study a hydrodynamic system modeling the flow of nematic liquid crystals. In three dimensions, we first establish local well-posedness of the initial-boundary value problem of the system. Then, we prove the existence of global strong solution to the system with small initial-boundary condition.
Xingxing Liu, Zhijun Qiao, Zhaoyang Yin
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
Weikui Ye, Zhaoyang Yin
In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation by constructing some special spaces $E^s_{p,r}$, which mix Lebesgue spaces and homogeneous Besov spaces. Then we present a global existence result and provide a sufficient condition for strong solutions to blow up in finite time for the equation. Finally, we give the ill-posedness and the unique continuation of the Hunter-Saxton equation.
Wenjie Deng, Zhaonan Luo, Zhaoyang Yin
In this paper, we are concerned with global strong solutions and large time behavior for some inviscid Oldroyd-B models. We first establish the energy estimate and B-K-M criterion for the 2-D co-rotation inviscid Oldroyd-B model. Then, we obtain global strong solutions with large data in Sobolev space by proving the boundedness of vorticity. As a corollary, we prove global existence of the corresponding inviscid Hooke model near equilibrium. Furthermore, we present global existence for the 2-D co-rotation inviscid Oldroyd-B model in critical Besov space by a refined estimate in Besov spaces with index $0$. Finally, we study large time behaviour for the noncorotation inviscid Oldroyd-B model. Applying the Fourier splitting method, we prove the $H^1$ decay rate for global strong solutions constructed by T. M. Elgindi and F. Rousset.
Wenjie Deng, Wei Luo, Zhaoyang Yin
In this paper, we mainly study the global strong solutions and its long time decay rates of all order spatial derivatives to a micro-macro model for compressible polymeric fluids with small initial data. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first prove that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for $d\geq2$. Moreover, for $d\geq3$, we also show a new critical Fourier estimation that allow us to give the long time decay rates of $L^2$ norm for all order spatial derivatives.
Jun Wang, Zhaoyang Yin
In this paper, we study the following nonlinear magnetic Schrödinger equation with logarithmic nonlinearity \begin{equation*} -(\nabla+iA(x))^2u+λV(x)u =|u|^{q-2}u+u\log |u|^2,\ u\in H^1(\mathbb{R}^N,\mathbb{C}), \end{equation*} where the magnetic potential $A \in L_{l o c}^2\left(\mathbb{R}^N, \mathbb{R}^N\right)$, $2<q<2^*,\ λ>0$ is a parameter and the nonnegative continuous function $V: \mathbb{R}^N \rightarrow \mathbb{R}$ has the deepening potential well. Using the variational methods, we obtain that the equation has at least $2^k-1$ multi-bump solutions when $λ>0$ is large enough.
Yingying Guo, Weikui Ye, Zhaoyang Yin
For the famous Camassa-Holm equation, the well-posedness in $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty)$ and the ill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $ p\in [1,\infty],\ r\in (1,\infty]$ had been studied in \cite{d1,d2,glmy,yyg}, that is to say, it only left an open problem in the critical case $B^{1}_{\infty,1}(\mathbb{R})$ proposed by Danchin in \cite{d1,d2}. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$. Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critial Besov spaces $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $ p\in [1,\infty]$ have been completed. Finally, since the norm inflation occurs by choosing an special initial data $u_0\in B^{1}_{\infty,1}(\mathbb{R})$ but $u^2_{0x}\notin B^{0}_{\infty,1}(\mathbb{R})$ (an example implies $B^{0}_{\infty,1}(\mathbb{R})$ is not a Banach algebra), we then prove that this condition is necessary. That is, if $u^2_{0x}\in B^{0}_{\infty,1}(\mathbb{R})$ holds, then the Camassa-Holm equation has a unique solution $u(t,x)\in \mathcal{C}_T(B^{1}_{\infty,1}(\mathbb{R}))\cap \mathcal{C}^{1}_T(B^{0}_{\infty,1}(\mathbb{R}))$ and the norm inflation will not occur.
Jinlu Li, Weipeng Zhu, Zhaoyang Yin
In this paper, we study the Cauchy problem for the following Hamilton-Jacobi equation \bbal\bca \pa_tu-\De u=|\na u|^2,\quad t>0, \ x\in \R^d,\\ u(0,x)=u_0, \quad \quad x\in \R^d. \eca\end{align*} We show that the solution map in Besov spaces $B^0_{\infty,q}(\R^d),1\leq q\leq \infty$ is discontinuous at origin. That is, we can construct a sequence initial data $\{u^N_0\}$ satisfying $||u^N_0||_{B^0_{\infty,q}(\R^d)}\rightarrow 0, \ N\rightarrow \infty$ such that the corresponding solution $\{u^N\}$ with $u^N(0)=u^N_0$ satisfies \bbal ||u^N||_{L^\infty_T(B^0_{\infty,q}(\R^d))}\geq c_0, \qquad \forall \ T>0, \quad N\gg 1, \end{align*} with a constant $c_0>0$ independent of $N$.
Zeng Zhang, Zhaoyang Yin
This paper is concerned with a multi-component Camassa-Holm system, which has been proven to be integrable and has peakon solutions. This system includes many one-component and two-component Camassa-Holm type systems as special cases. In this paper, we first establish the local well-posedness and a continuation criterion for the system, then we present several global existence or blow-up results for two important integrable two-component subsystems. Our obtained results cover and improve recent results in \cite{Gui,yan}.
Chunxia Guan, Xi Tu, Zhaoyang Yin
In this paper, we prove the existence of a global entropy weak solution $u\in H^1(\mathbb{R})$ and $\partial_{x}u\in L^1(\mathbb{R})\cap BV(\mathbb{R})$ for the Cauchy problem of a generalized Camassa-Holm equation by the viscous approximation method.
Wenke Tan, Zhaoyang Yin
In this paper, we consider Cauchy problem of simplified Ericksen-Leslie system in dimension three. We establish the unique existence of global smooth solution under some nonlinear conditions on initial data. However, we do not need small conditions on initial data.
Wei Luo, Zhaoyang Yin
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.
Xiaoping Zhai, Zhaoyang Yin
The present paper is dedicated to the global well-posedness for the 3D inhomogeneous incompressible Navier-Stokes equations, in critical Besov spaces without smallness assumption on the variation of the density. We aim at extending the work by Abidi, Gui and Zhang (Arch. Ration. Mech. Anal. 204 (1):189--230, 2012, and J. Math. Pures Appl. 100 (1):166--203, 2013) to a more lower regularity index about the initial velocity. The key to that improvement is a new a priori estimate for an elliptic equation with nonconstant coefficients in Besov spaces which have the same degree as $L^2$ in $\mathbb{R}^3$. Finally, we also generalize our well-posedness result to the inhomogeneous incompressible MHD equations.
Xi Tu, Zhaoyang Yin
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.