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Zhaoyang Yin

Publications and source records attributed to Zhaoyang Yin.

At least 37 records · Page 2Linked to original sources
arXiv2025

Existence and dependency results for coupled Schrödinger equations with critical exponent on waveguide manifold

We study the coupled Schrödinger equations with critical exponent on $\mathbb{R}^3 \times \mathbb{T}$. With the help of scaling argument and semivirial-vanishing technology, we obtain the existence and $y$-dependence of solution, the tori can be generalized to $1$-dimensional compact Riemannian manifold. Moreover, the conclusion of this paper can be extended to systems with any number of components.

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arXiv2025

Sharp non-uniqueness for the Boussinesq equation with fractional dissipation

This paper focuses on the $d$-dimensional ($d\geq2$) Boussinesq equation with fractional dissipation $(-Δ)^α$ on the torus. We show that the uniqueness property breaks down within the function space $L^p_tL^\infty_x$ for any $p<\frac{2α}{2α-1}$ when $1\leqα<\frac{d+1}{2}$ and the function space $L^\frac{2α}{2α-1}_tL^q_x$ for any $q<\infty$ when $1<α<\frac{d+1}{2}$. Moreover, the weak solutions we construct are smooth outside a set of singular times with Hausdorff dimension arbitrarily small. This result is sharp, as weak-strong uniqueness holds in the space $L^{\frac{2α}{2α-1}}_TL^\infty_x$.

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arXiv2025

The local well-posedness, blow-up phenomena and ill-posedness of a new fifth-order Camassa-Holm type equation

In this paper, we study a new fifth-order Camassa-Holm type equation derived by Li \cite{Li.Z}. We firstly establish the local well-posedness in the sense of Hadamard for the Cauchy problem of the new fifth-order Camassa-Holm type equation in Besov spaces. Secondly, we obtain blow-up criteria. Building upon this, by utilizing the conservation laws and establishing local boundedness, we derive a blow-up result that precisely determines the blow-up time. Finally, the ill-posedness of the new fifth-order Camassa-Holm type equation in the critical Sobolev space $H^{\frac{1}{2}}$ is established via a norm inflation argument.

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arXiv2025

The local well-posedness, global existence and ill-posedness for the fifth order Camassa-Holm model

In this paper, we consider the fifth order Camassa-Holm model. Firstly, we improve the local well-posedness results in \cite{TangLiu2015,FOCH2021}. Secondly, we give the blow up criteria and conditions for global existence. Finally, when $b=\frac 53$ in the model, we obtain the ill-posedness in $B^1_{\infty,1}$ and $B^{\frac 32}_{2,q}$ with $q\in(1,+\infty]$ in the sense of norm inflation.

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arXiv2025

Hölder continuous weak solutions of the 3D Boussinesq equation with thermal diffusion

In this paper, we show the existence of Hölder continuous periodic weak solutions of the 3D Boussinesq equation with thermal diffusion, which apprroximate the Onsager's critical spatial regularity and satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,T]\rightarrow \mathbb{R}_+$ and $β\in (0, \frac{1}{3})$, there exist $v\in C^β([0,T]\times {\mathbb{T} }^3)$ and $ θ\in C_t^{1,\fracβ{2}}C_x^{2,β}([0,T]\times {\mathbb{T} }^3)$ which solve (\ref{e:boussinesq equation}) in the sense of distribution and satisfy \begin{align} e(t)=\int_{{{\mathbb{T} }^3}}|v(t,x)|^2dx, \quad \forall t\in [0,T].\nonumber \end{align}

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arXiv2024

Orbital stability of normalized ground states for critical Choquard equation with potential

In this paper, we study the existence of ground state standing waves and orbital stability, of prescribed mass, for the nonlinear critical Choquard equation \begin{equation*} \left\{\begin{array}{l} i \partial_t u+Δu -V(x)u+(I_α\ast|u|^{q})|u|^{q-2}u+(I_α\ast|u|^{2_α^*})|u|^{2_α^*-2}u=0,\ (x, t) \in \mathbb{R}^d \times \mathbb{R}, \\ \left.u\right|_{t=0}=φ\in H ^1(\mathbb{R}^d), \end{array}\right. \end{equation*} where $I_α$ is a Riesz potential of order $α\in(0,d),\ d\geq3,\ 2_α^*=\frac{2d-α}{d-2}$ is the upper critical exponent due to Hardy-Littlewood-Sobolev inequality, $\frac{2d-α}{d}<q<\frac{2d-α+2}{d}$. Under appropriate potential conditions, we obtain new Strichartz estimates and construct the new space to get orbital stability of normalized ground state. To our best knowledge, this is the first orbital stability result for this model. Our method is also applicable to other mixed nonlinear equations with potential.

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arXiv2024

Almost sure well-posedness and orbital stability for Schrödinger equation with potential

In this paper, we study the almost sure well-posedness theory and orbital stability for the nonlinear Schrödinger equation with potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+Δu-V(x)u+|u|^{2}u=0,\ (x, t) \in \mathbb{R}^4 \times \mathbb{R}, \\ \left.u\right|_{t=0}=f \in H ^s(\mathbb{R}^4), \end{array}\right. \end{equation*} where $\frac{1}{3}<s<1$ and $V(x):\mathbb{R}^4\rightarrow \mathbb{R}$ satisfies appropriate conditions. The main idea in the proofs is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing and maximal function spaces. To our best knowledge, this is the first orbital stability result for this model.

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arXiv2024

Normalized solutions and stability for biharmonic Schrödinger equation with potential on waveguide manifold

In this paper, we study the following biharmonic Schrödinger equation with potential and mixed nonlinearities \begin{equation*} \left\{\begin{array}{ll}Δ^2 u +V(x,y)u+λu =μ|u|^{p-2}u+|u|^{q-2}u,\ (x, y) \in Ω_r \times \mathbb{T}^n, \\ \int_{Ω_r\times\mathbb{T}^n}u^2dxdy=Θ,\end{array} \right. \end{equation*} where $Ω_r \subset \mathbb{R}^d$ is an open bounded convex domain, $r>0$ is large and $μ\in\mathbb{R}$. The exponents satisfy $2<p<2+\frac{8}{d+n}<q<4^*=\frac{2(d+n)}{d+n-4}$, so that the nonlinearity is a combination of a mass subcritical and a mass supercritical term. Under some assumptions on $V(x,y)$ and $μ$, we obtain the several existence results on waveguide manifold. Moreover, we also consider the orbital stability of the solution.

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arXiv2024

Global regularity and optimal decay estimates of large solutions to the compressible FENE system

In this paper, we are concerned with the compressible FENE dumbbell model. By virtue of the dissipative structure and the interpolation method, we firstly prove global regularity in $H^2$ framework for the compressible FENE system with some large data. Then, we obtain optimal decay estimates of large solutions in $H^1$ and remove the smallness assumption of low frequencies by virtue of the Fourier splitting method and the Littlewood-Paley decomposition theory. Furthermore, we establish optimal decay rate for the highest derivative of the solutions by a different method combining time frequency decomposition and the time weighted energy estimate. These obtained results generalize and cover the classical results of the incompressible FENE dumbbell model.

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arXiv2024

The local well-posedness, blow-up and global solution of a new integrable system in Besov spaces

In this paper, we first establish the local well-posednesss for the Cauchy problem of a $N$-peakon system in the sense of Hadamard in both critical Besov spaces and supercritical Besov spaces. Second, we gain a blow-up criterion. According to blow-up criterion wemreach a precise blow-up criterion. Under a sign condition, we reach the existence of global solution. Finally, based on the first-order difference method, we give a simulation example of the blow-up of the equation and the properties of the global solution.

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arXiv2024

Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with indefinite potential in the radial case

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schrödinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+Δu-V(x)u +|u|^{\frac{4}{N-2}}u=0,\ (x, t) \in \mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H ^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. Moreover, we get a positive ground state solution which is radially symmetric by using variational methods. This paper extends the results of \cite{KCEMF2006}(Invent. Math) to the potential equation and develops the recent conclusions.

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arXiv2024

The uniform existence time and Zero-Alpha limit problem of the Euler-Poincaré equations

We consider the Cauchy problem of the Euler-Poincaré equations in $\mathbb{R}^d$ with a varying dispersion parameter $α$. Based on the convex entropy structure and the modified commutator estimates, we have proved that the Euler-Poincaré equations have a uniform existence time with respect to $α$ in Sobolev spaces $H^s.$ Combined with the Bona-Simth method, we obtain convergence of the solutions to the Euler-Poincaré equations as $α\to 0$ in the same space where the initial data are located.

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arXiv2024

Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger equation with general nonlinearity in the radial case

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schrödinger equation with general nonlinearity \begin{equation*} \left\{\begin{array}{l} i \partial_t u+Δu + f(u)=0,\ (x, t) \in \mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H ^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where $f:\mathbb{C}\rightarrow \mathbb{C}$ satisfies Sobolev critical growth condition. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. This paper generalizes the conclusions in \cite{KCEMF2006}(Invent. Math).

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arXiv2024

The well-posedness and blow up phenomenon for a Tsunamis model with time-fractional derivative

This paper is concerned with the well-posedness of a time-fractional shallow-water equations, which has received little attention. In the realm of fractional calculus, numerous types of fractional derivatives have been explored in the literature. Among these, one of the most notable and well-structured ones is the conformable fractional derivative. In this paper, we delve into the local well-posedness of the fractional tsunami shallow-water mathematical model in the critical Besov space $B^{\frac{3}{2}}_{2,1}$. Under some symmetric and sign conditions, we show that the strong solution will blow up in finite time.

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arXiv2024

Multi-bump solutions for the nonlinear magnetic Schrödinger equation with logarithmic nonlinearity

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 $A \in L_{l o c}^2\left(\mathbb{R}^N, \mathbb{R}^N\right)$, $2 0$ 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.

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arXiv2023

Global existence and optimal decay rate of weak solutions to the co-rotation Hooke dumbbell model

In this paper, we mainly study global existence and optimal $L^2$ decay rate of weak solutions to the co-rotation Hooke dumbbell model. This micro-macro model is a coupling of the Navier-Stokes equation with a nonlinear Fokker-Planck equation. Based on the defect measure propagation method, we prove that the co-rotation Hooke dumbbell model admits a global weak solution provided the initial data under different integrable conditions. Moreover, we obtain optimal time decay rate in $L^2$ for the weak solutions obtained by the Fourier splitting method.

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