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Zhaoyang Yin

Publications and source records attributed to Zhaoyang Yin.

At least 91 records · Page 5Linked to original sources
arXiv2018

Local bifurcation of steady almost periodic water waves with constant vorticity

In this paper we mainly investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem with some unknown functions in the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for a almost periodic function of one variable. The bifurcation theory ensures us to obtain a existence result. Our existence result generalizes and covers the recent result in \cite{Constantin2011v}. Moreover, our result implies a non-uniqueness result at the same bifurcation point.

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arXiv2018

Local well-posedness of the incompressible Euler equations in $B^1_{\infty,1}$ and the inviscid limit of the Navier-Stokes equations

We prove the inviscid limit of the incompressible Navier-Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier-Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona-Smith type method in the $L^p$ setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space $B^{\frac dp+1}_{p,1}(\mathbb{R}^d)$, $1\leq p\leq \infty$, $d\geq 2$, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in \cite{BL,BL1} and by Misiołek and Yoneda in \cite{MY,MY2, MY3}.

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arXiv2018

Global existence and well-posedness for the Doi-Edwards polymer model

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension $d\geq2$. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contain a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result.

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arXiv2017

The $L^2$ decay for the 2D co-rotation FENE dumbbell model of polymeric flows

In this paper we mainly study the long time behaviour of solutions to the finite extensible nonlinear elastic (FENE) dumbbell model with dimension two in the co-rotation case. Firstly, we obtain the $L^2$ decay rate of the velocity of the 2D co-rotation FENE model is $(1+t)^{-\frac{1}{2}}$ with small data. Then, by virtue of the Littlewood-Paley theory, we can remove the small condition. Our obtained sharp result improves considerably the recent results in \cite{Luo-Yin,Schonbek}.

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arXiv2017

Ill-posedness for the Hamilton-Jacobi equation in Besov spaces $B^0_{\infty,q}$

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 $N$.

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arXiv2017

A note on the inviscid limit of the incompressible MHD equations

In this paper, we prove that as the viscosity and resistivity go to zero, the solution of the Cauchy problem for the incompressible MHD equations converges to the solution of the ideal MHD equations in the same topology with the initial data. Our proof mainly depends on the method introduced by the paper \cite{G.L.Y} and the constructions of the incompressible MHD equations.

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arXiv2017

Global Gevrey regularity and analyticity of a two-component shallow water system with Higher-order inertia operators

In this paper, we mainly consider the Gevrey regularity and analyticity of the solution to a generalized two-component shallow water wave system with higher-order inertia operators, namely, $m=(1-\partial_x^2)^su$ with $s>1$. Firstly, we obtain the Gevrey regularity and analyticity for a short time. Secondly, we show the continuity of the data-to-solution map. Finally, we prove the global Gevrey regularity and analyticity in time.

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arXiv2016

Global conservative solution for the periodic $μ$-Camassa-Holm equation

In this paper we mainly investigate the periodic $μ$-Camassa-Holm equation. We show the existence of global conservative solutions to the Cauchy problem of the periodic $μ$-Camassa-Holm equation. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. Our solutions depend continuously on the initial data and has a semigroup property.

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arXiv2016

Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations

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.

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arXiv2016

Global Weak Solution for a generalized Camassa-Holm equation

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.

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arXiv2015

Gevrey regularity and analyticity for Camassa-Holm type systems

In this paper we mainly investigate the Cauchy problem of some Camassa-Holm type systems. By constructing a new auxiliary function, we present a generalized Ovsyannikov theorem. By using this theorem and the basic properties of Sobolev-Gevrey spaces, we prove the Gevrey regularity and analyticity of these systems. Moreover, we obtain a lower bound of the lifespan and the continuity of the data-to-solution map.

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arXiv2015

Blow-up phenomena and local well-posedness for a generalized Camassa-Holm equation

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 simply 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.

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arXiv2015

Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space

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.

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