arXiv subjects
Publications and source records attributed to 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.
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}.
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}.
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}
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 $s_2$.
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].
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 $ν^\frac{1}{2}$.
In this paper we show the Gevrey regularizing effect of solutions to the non-cutoff spatially homogeneous and inhomogeneous Boltzmann equation for a particular soft potential with critical singularity s=1/2.
In this paper we study the Gevrey regularity for the weak solutions to the Cauchy problem of the non-cutoff spatially homogeneous Botlzmann equation for the Maxwellian molecules model with the singularity exponent $s\in (0,1)$. We establish that any weak solution belongs to the Gevrey spaces for any positive time.
In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order $1/(2s)$ in the velocity variable $v$ and order 1 in the space variable $x$. The strategy relies on our recent results for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023). Rather, we need much more intricate analysis additionally in order to handle with the coupling of the double variables. Combining with the previous result mentioned above, it gives a whole characterization of the Gevrey regularity of the particular kind of solutions to the non-cutoff Boltzmann.
In this paper we study the Gevrey smoothing effect of solutions to the non-cutoff spatially homogeneous and inhomogeneous Boltzmann equation for soft potential. We consider the mild singularity case $s<1/2$ as we did in the previous work for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023) and for spatially inhomogeneous case (arXiv:1304.2971), and try to extend the range of $γ$. We derive a new coercivity estimate for collision operator, using which we can obtain the Gevrey regularity for $γ\in (-5/2,0)$ improving the previous assumption $γ\in (-1-2s,0)$. Besides, we consider $γ$ and $s$ separately instead of viewing $γ+2s$ as one untied quantity.
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 $χ^{-1}$. Then we show that if the $χ^{-1}$ norm of the initial data is smaller than C$ν$ in the GNS system where $ν$ 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 $χ^{-1}$ norm of the initial data is less than $ν.$ Our obtained results cover and improve recent results in \cite{Zhen Lei,wu}.
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.
This paper is contributed to study the Cauchy problem of a new integrable two-component system with peaked soliton (peakon) and weak kink solutions. We first establish the local well-posedness result for the Cauchy problem in Besov spaces, and then present a precise blow-up scenario and a new blow-up result for strong solutions to the system.
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.
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.
In this paper, we study the Cauchy problem of a weakly dissipative $μ$HS equation. We first establish the local well-posedness for the weakly dissipative $μ$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.
This paper is concerned with global existence of weak solutions for a weakly dissipative $μ$HS equation by using smooth approximate to initial data and Helly$^{,}$s theorem.