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Zhaoyang Yin

Publications and source records attributed to Zhaoyang Yin.

At least 73 records · Page 4Linked to original sources
arXiv2021

Global Regularity and instability for the incompressible non-viscous Oldroyd-B model

In this paper, we consider the 2-dimensional non-viscous Oldroyd-B model. In the case of the ratio equal 1~($α=0$), it is a difficult case since the velocity field $u(t,x)$ is no longer decay. Fortunately, by {observing the exponential decay} of the stress tensor $τ(t,x)$, we succeeded in proving the global existence for this system with some large initial data. Moreover, we give an unsteady result: when the ratio is close to 1~($a\rightarrow 0$), the system is not steady for large time. This implies an interesting physical phenomenon that the term $a\mathbb{D}u$ is a bridge between the transformation of kinetic energy $u$ and elastic potential energy $τ$, but this process is transient for large time, which leads the instability.

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arXiv2021

Well-posedness and global solutions to the higher order Camassa-Holm equations with fractional inertia operator in Besov space

In this paper, we study well-posedness and the global solutions to the higher-order Camassa-Holm equations with fractional inertia operator in Besov space. When $a\in[\frac{1}{2},1),$ we prove the existence of the solutions in space $B^s_{p,1}(\mathbb R)$ with $s\geq 1+\frac{1}{p}$ and $p <\frac{1}{a-\frac{1}{2}}$, the existence and uniqueness of the solutions in space $B^s_{p,1}(\mathbb R)$ with $s\geq 1+2a-\min\{\frac{1}{p},\frac{1}{p'}\},$ and the local well-posedness in space $B^s_{p,1}(\mathbb R)$ with $s> 1+2a-\min\{\frac{1}{p},\frac{1}{p'}\}$. When $a>1,$ we obtain the existence of the solutions in space $B^s_{p,1}(\mathbb R)$ with $s\geq a+\max\{\frac{1}{p},\frac{1}{2}\}$ and the local well-posedness in space $B^s_{p,1}(\mathbb R)$ with $s\geq 1+a+\max\{\frac{1}{p},\frac{1}{2}\}$. Moreover, we obtain two results about the global solutions.

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arXiv2021

Large time behavior of global strong solutions to the 2-D compressible FENE dumbbell model

In this paper, we mainly study large time behavior of the strong solutions to the 2-D compressible finite extensible nonlinear elastic (FENE) dumbbell model. The Fourier splitting method yields that the $L^2$ decay rate is $\ln^{-l}(e+t)$ for any $l\in\mathbb{N}$. By virtue of the time weighted energy estimate, we can improve the decay rate to $(1+t)^{-\frac{1}{4}}$. Under the low-frequency condition and by the Littlewood-Paley theory, we show that the solutions belong to some Besov space with negative index and obtain the optimal $L^2$ decay rate.

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arXiv2021

Global strong solutions and optimal $L^2$ decay to the compressible FENE dumbbell model

In this paper, we are concerned with the global well-posedness and $L^2$ decay rate for the strong solutions of the compressible finite extensible nonlinear elastic (FENE) dumbbell model. For $d\geq 2$, we prove that the compressible FENE dumbbell model admits a unique global strong solution provided the initial data are close to equilibrium state. Moreover, by the Littlewood-Paley decomposition theory and the Fourier splitting method, we show optimal $L^2$ decay rate of global strong solutions for $d\geq 3$.

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arXiv2021

Global well-posedness for the non-viscous MHD equations with magnetic diffusion in critical Besov spaces

In this paper, we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion. We first establish the local well-posedness (existence,~uniqueness and continuous dependence) with initial data $(u_0,b_0)$ in critical Besov spaces $ {B}^{\frac{d}{p}+1}_{p,1}\times{B}^{\frac{d}{p}}_{p,1}$ with $1\leq p\leq\infty$, and give a lifespan $T$ of the solution which depends on the norm of the Littlewood-Paley decomposition of the initial data. Then, we prove the global existence in critical Besov spaces. In particular, the results of global existence also hold in Sobolev space $ C([0,\infty); {H}^{s}(\mathbb{S}^2))\times \Big(C([0,\infty);{H}^{s-1}(\mathbb{S}^2))\cap L^2\big([0,\infty);{H}^{s}(\mathbb{S}^2)\big)\Big)$ with $s>2$, when the initial data satisfies $\int_{\mathbb{S}^2}b_0dx=0$ and $\|u_0\|_{B^1_{\infty,1}(\mathbb{S}^2)}+\|b_0\|_{{B}^{0}_{\infty,1}(\mathbb{S}^2)}\leq ε$. It's worth noting that our results imply some large and low regularity initial data for the global existence, which improves considerably the recent results in \cite{weishen}.

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arXiv2021

The continuous dependence for the Navier-Stokes equations in $\dot{B}^{\frac{d}{p}-1}_{p,r}$

In this paper, we mainly investigate the Cauchy problem for the incompressible Navier-Stokes equations in homogeneous Besov spaces $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq p<\infty,\ 1\leq r\leq \infty, \ d\geq 2$. Firstly, we prove the local existence of the solution and give a lower bound of the lifespan $T$ of the solution. The lifespan depends on the Littlewood-Paley decomposition of the initial data, that is $\dotΔ_j u_0$. Secondly, if the initial data $u^n_0\rightarrow u_0$ in $\dot{B}^{\frac{d}{p}-1}_{p,r}$, then the corresponding lifespan $T_n\rightarrow T$. Thirdly, we prove that the data-to-solutions map is continuous in $\dot{B}^{\frac{d}{p}-1}_{p,r}$. Therefore, the Cauchy problem of the Navier-Stokes equations is locally well-posed in the critical Besov spaces in the Hadamard sense. Moreover, we also obtain well-posedness and weak-strong uniqueness results in $L^{\infty}L^2\cap L^{2}\dot{H}^1$.

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arXiv2020

On the Cauchy problem for the Hunter-Saxton equation on the line

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.

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arXiv2020

The existence and uniqueness of global admissible conservative weak solution for the periodic single-cycle pulse equation

This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.

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arXiv2020

The estimate of lifespan and local well-posedness for the non-resistive MHD equations in homogeneous Besov spaces

In this paper, we mainly investigate the Cauchy problem of the non-resistive MHD equation. We first establish the local existence in the homogeneous Besov space $\dot{B}^{\frac{d}{p}-1}_{p,1}\times \dot{B}^{\frac{d}{p}}_{p,1}$ with $p<\infty$, and give a lifespan $T$ of the solution which depends on the norm of the Littlewood-Paley decomposition of the initial data. Then, we prove that if the initial data $(u^n_0,b^n_0)\rightarrow (u_0,b_0)$ in $\dot{B}^{\frac{d}{p}-1}_{p,1}\times \dot{B}^{\frac{d}{p}}_{p,1}$, then the corresponding existence times $T_n\rightarrow T$, which implies that they have a common lower bound of the lifespan. Finally, we prove that the data-to-solutions map depends continuously on the initial data when $p\leq 2d$. Therefore the non-resistive MHD equation is local well-posedness in the homogeneous Besov space in the Hadamard sense. Our obtained result improves considerably the recent results in \cite{Li1,chemin1,Feffer2}.

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arXiv2020

Large time behavior to the FENE dumbbell model of polymeric flows near equilibrium

In this paper we mainly study large time behavior for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. There is a lot results about the $L^2$ decay rate of the co-rotation model. In this paper, we consider the general case. We prove that the optimal $L^2$ decay rate of the velocity is $(1+t)^{-\frac{d}{4}}$ with $d\geq 2$. This result improves the previous result in \cite{Luo-Yin}.

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arXiv2020

Vanishing viscosity limit to the FENE dumbbell model of polymeric flows

In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also.

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arXiv2019

Remarks on the global large solution to the three-dimensional incompressible Navier-Stokes equations

In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations. That is, we prove that there exist two positive constants $c_0,C_0$ such that if \begin{equation*} \|u_0^1+u^2_0,u^3_0\|_{\dot{B}_{p,1}^{-1+\frac{3}{p}}} \|u^1_0,u^2_0\|_{\dot{B}_{p,1}^{-1+\frac{3}{p}}} \exp\{C_0 (\|u_0\|^{2}_{\dot{B}_{\infty,2}^{-1}}+\|u_0\|_{\dot{B}_{\infty,1}^{-1}})\} \leq c_0, \end{equation*} then \eqref{NS} has a unique global solution. As an application we construct two family of smooth solutions to the Navier-Stokes equations whose $\B^{-1}_{\infty,\infty}(\mathbb{R}^3)$ norm can be arbitrarily large.

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arXiv2019

Global large solution to the compressible Navier-Stokes equations in critical Besov space $\dot{B}^{-1}_{\infty,\infty}$

In this paper, we construct a class of global large solution to the compressible Navier-Stokes equations in the whole space $\R^d$. Precisely speaking, our choice of special initial data whose $\dot{B}^{-1}_{\infty,\infty}$ norm can be arbitrarily large, namely, $||u_0||_{\dot{B}^{-1}_{\infty,\infty}}\gg 1$, allows to give rise to global-in-time solution to the compressible Navier-Stokes equations.

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arXiv2019

Global existence and blow-up phenomena for two-component Degasperis-Procesi system and two-component b-family system

This paper is concerned with global existence and blow-up phenomena for two-component Degasperis-Procesi system and two-component b-family system. The strategy relies on our observation on new conservative quantities of these systems. Several new global existence results and a new blowup result of strong solutions to the two-component Degasperis- Procesi system and the two-component b-family system are presented by using these new conservative quantities.

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arXiv2018

Non-uniform continuous dependence and continuous dependence for the non-resistive MHD equations

In this paper, we establish the continuous dependence for the non-resistive MHD equations in Sobolev spaces. Our obtained result fills considerably the recent result [C. Fefferman, D. McCormick, J. Robinson and J. Rodrigo, Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal. 267 (2014), 1035--1056] for the continuous dependence on initial data. We also show that this result is optimal. More precisely, we proved that the data-to-solution map is not uniform continuous dependence in Sobolev spaces.

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arXiv2018

Blow-up phenomenon, ill-posedness and peakon solutions for the periodic Euler-Poincaré equations

In this paper we mainly investigate the initial value problem of the periodic Euler-Poincaré equations. We first present a new blow-up result to the system for a special class of smooth initial data by using the rotational invariant properties of the system. Then, we prove that the periodic Euler-Poincaré equations is ill-posed in critical Besov spaces by a contradiction argument. Finally, we verify the system possesses a class of peakon solutions in the sense of distributions.

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arXiv2018

Ill-posedness of the Camassa-Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (\cite{Danchin2,Byers,HHK}).

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