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Zhaoyang Yin

Publications and source records attributed to Zhaoyang Yin.

At least 55 records · Page 3Linked to original sources
arXiv2023

Normalized solutions for Schödinger equations with potential and general nonlinearities involving critical case on large convex domains

In this paper, we study the following Schrödinger equations with potentials and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -Δu+V(x)u+λu=|u|^{q-2}u+βf(u), \\ & \int |u|^2dx=Θ, \end{aligned} \right. \end{equation*} both on $\mathbb{R}^N$ as well as on domains $r Ω$ where $Ω\subset \mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponent satisfies $2+\frac{4}{N}\leq q\leq2^*=\frac{2 N}{N-2}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $L^2$-subcritical or $L^2$-critical growth. This paper generalizes the conclusion of Bartsch et al. in \cite{TBAQ2023}(2023, arXiv preprint). Moreover, we consider the Sobolev critical case and $L^2$-critical case of the above problem.

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arXiv2023

Global regularity and large time behavior for some inviscid Oldroyd-B models in $\mathbb{R}^2$

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.

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arXiv2023

Global existence and optimal decay rate of weak solutions to some inviscid Oldroyd-B models

This paper is devoted to global existence and optimal decay rate of weak solutions to some inviscid Oldroyd-B models with center diffusion. By virtue of the properties of Calderon-Zygmund operator and the Littlewood-Paley decomposition theory, we firstly prove that the 2-D co-rotation inviscid Oldroyd-B model admits global weak solutions with some large data under different integrability conditions. Furthermore, we prove the energy conservation of weak solutions for the co-rotation case. These obtained results generalize and cover the classical results for the Euler equation. Moreover, we establish global weak solutions with small data for the 2-D noncorotation inviscid Oldroyd-B model without damping. Finally, we prove optimal decay rate of global weak solutions for the noncorotation case by the improved Fourier splitting method.

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arXiv2023

Global strong solutions and large time behavior to the compressible co-rotation FENE dumbbell model of polymeric flows near equilibrium

In this paper, we mainly study global well-posedness and optimal decay rate for the strong solutions of the compressible co-rotation finite extensible nonlinear elastic (FENE) dumbbell model. This model is a coupling of the isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first prove that the FENE dumbbell model admits a unique global strong solution provided the initial data are close to equilibrium state for $d\geq 2$. Moreover, for $d\geq3$, we show that optimal decay rates of global strong solutions by the linear spectral theory and a more precise Hardy type inequality.

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arXiv2023

Optimal decay rate for the 2-D compressible Oldroyd-B and Hall-MHD model

In this paper, we are concerned with long time behavior of the strong solutions to the 2-D compressible Oldroyd-B and Hall-MHD model. By virtue of the improved Fourier splitting method and the time weighted energy estimate, we obtain the $L^2$ decay rate $(1+t)^{-\frac{1}{4}}$. According to the Littlewood-Paley theory, we prove that the solutions belong to the critical Besov space with negative index. Finally, we show optimal decay rate in $H^2$-framework without the smallness restriction of low frequencies.

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arXiv2023

Optimal decay rate for the generalized Oldroyd-B model with only stress tensor diffusion in $\mathbb{R}^2$

In this paper, we are concerned with optimal decay rate for the 2-D generalized Oldroyd-B model with only stress tensor diffusion $(-Δ)^βτ$. In the case $β=1$, we first establish optimal decay rate in $H^1$ framework and remove the smallness assumption of low frequencies by virtue of the Fourier splitting method and the Littlewood-Paley decomposition theory. Furthermore, we prove optimal decay rate for the highest derivative of the solution by a different method combining time frequency decomposition and the time weighted energy estimate. In the case $\frac 1 2\leq β<1$, we study optimal decay rate for the highest derivative of the solution by the improved Fourier splitting method.

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arXiv2023

Large time behavior to a 2D micro-macro model for compressible polymeric fluids near equilibrium

In this paper, we mainly study the large time behavior to a 2D 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. Firstly the Fourier splitting method yields that the logarithmic decay rate. 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 spaces with negative index and obtain the optimal $L^2$ decay rate. Finally, we obtain the $\dot{H}^s$ decay rate by establishing a new Fourier splitting estimate.

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arXiv2022

Global strong solutions and large time behavior to a micro-macro model for compressible polymeric fluids near equilibrium

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.

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arXiv2022

Global Gevery regulartiy and analyticity of a weakly dissipative Camassa-Holm equation

This work is concerned with the Gevrey regularity and analyticity of the solution to a weakly dissipative Camassa-Holm system. We first demonstrate the local Gevery regularity and analyticity of this equation. Then, we disscuss the continuity of the data-to-solution map. Finally, we obtain the global Gevery regularity of this system in Gevery class $G_σ$ with $σ\geq 1$ in time.

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arXiv2022

Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa-Holm equation in time weighted $H^1(\mathbb{R})$ space

In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by introducing some new variables, and present the globally conservative solutions of this equation in time weighted $H^1$ space. Second, we show that the peakon solutions are conservative weak solutions in $H^1.$ Finally, given a conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system having unique solutions. In turn, we get the uniqueness of the conservative solution in the original variables.

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arXiv2022

On the Cauchy problem for a weakly dissipative Camassa-Holm equation in critical Besov spaces

In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces $B^{s}_{p,r}$ with $s>1+\frac 1 p$ and $s=1+\frac 1 p , r=1,p\in [1,\infty).$ Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces.

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arXiv2022

A new result for the local well-posedness of the Camassa-Holm type equations in critial Besov spaces $B^{1+\frac{1}{p}}_{p,1},1\leq p<+\infty$

For the famous Camassa-Holm equation, the well-posedness in $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p\leq2$ and the ill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $1\leq p\leq+\infty,\ 1<r\leq+\infty$ had been studied in \cite{d1,d2,glmy}. That is to say, it left an open problem in the critical case $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $2<p\leq+\infty$ proposed by Danchin in \cite{d1,d2}. In this paper, we solve this problem. The main difficulty is to prove the uniqueness, which usually needs to use the Moser-type inequality, resulting in the index $p$ belongs to $[1,2]$. To overcome the difficulty, inspired by Linares, Ponce and Thomas \cite{lps}, we combine the Lagrange coordinate transformation and small time conditions to avoid using the Moser-type inequality. As a result, we obtain the local well-posedness for the Camassa-Holm equation in critical Besov spaces $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p<+\infty$. It is worth mentioning that our method is suitable for many Camassa-Holm type equations such as the Novikov equation and the two-component Camassa-Holm system, which can also improve their index on the local well-posedness.

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arXiv2022

Ill-posedness for the Cauchy problem of the Camassa-Holm equation in $B^{1}_{\infty,1}(\mathbb{R})$

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.

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arXiv2022

Local well-posedness and global existence for the Popowicz system

Popowicz system, as the interacting system of Camassa-Holm and Degasperis-Procesi equations, has attracted some attention in recent years. In this paper, we first study the local well-posedness for the cauchy problem of Popowicz system in nonhomogeneous Besov spaces $B^s_{p,r}\times B^s_{p,r}$ with $s> \max\{2, \frac{1}{p}+\frac{3}{2}\}$ or $(s=2, 2\leq p \leq \infty, 1\leq r\leq 2)$. Moreover, a new blow-up criterion and global existence with different initial values are obtained.

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arXiv2022

A new result for the local well-posedness of the generalized Camassa-Holm equations in critial Besov spaces $B^{\frac{1}{p}}_{p,1},1\leq p<+\infty$

This paper is devoted to studying the local well-posedness (existence,uniqueness and continuous dependence) for the generalized Camassa-Holm equations in critial Besov spaces $B^{\frac{1}{p}}_{p,1}$ with $1\leq p<+\infty$, which improves the previous index $s> \max\{\frac{1}{2},\frac{1}{p}\}$ or $s=\frac{1}{p},\ p\in[1,2],\ r=1$ in \cite{linb,tu-yin4}. The main difficulty is to prove the uniqueness, which need to use the Moser-type inequality. To overcome the difficulty, we use the Lagrange coordinate transformation to obtain the uniqueness.

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