arXiv subjects

Zhaoyang Yin

Publications and source records attributed to Zhaoyang Yin.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness and regularity for the Fractional Rough Burgers equation in Sobolev spaces with an application

In this paper, we study the well-posedness of Fractional Rough Burgers equation driven by space-time white noise in $H^s$ space. For the higher dissipation $γ\in(\frac{4}{3},2]$, we establish local well-posedness. Global well-posedness is further obtained when $γ$ is restricted to the interval $(\frac{8}{5}, 2]$. For the lower dissipation $γ\in(\frac{5}{4},\frac{4}{3}]$, we establish the para-controlled solution in $\mathcal C^{s}\cap H^s$.

math.AP↗

arXiv2026

A Generalized Framework for Singular Fractional Burgers Equations with Stochastic Forcing

In this paper, we investigate the regularization effect of fractional stochastic forcing on Burgers-type equations with fractional dissipation, with an application to the Degasperis--Procesi (DP) equation. In particular, we consider the perturbation induced by the singular noise $|D|^{1/2}ξ$ and establish local well-posedness in the negative Sobolev space $H^{-1/4+δ}$ for some small $δ>0$. Due to the singular nature of the nonlinear interactions, classical solution theories cannot be directly applied. Inspired by the framework developed in \cite{hairer2013solving,gubinelli2017kpz}, we introduce a generalized solution theory based on an enhanced structure and derive the effective equations satisfied by these generalized solutions. Our main contribution is the establishment of a general framework for describing singular PDEs driven by rough data. Moreover, we prove the convergence of the associated non-Gaussian rough structures in the fractional dissipation setting. As an application, we apply this framework to the stochastic Degasperis--Procesi equation and obtain its local well-posedness in the low-regularity regime.

math.AP↗

arXiv2026

Strong ill-posedness for the MHD system in the supercritical regime: inviscid and viscous

This paper is concerned with the Cauchy problem for the 3D incompressible magnetohydrodynamic (MHD) equations in supercritical Sobolev spaces. It is well known that the system is locally well-posed in subcritical Sobolev spaces, whereas the supercritical regime remains largely open. In this work, we establish norm inflation for the incompressible MHD equations, both with and without Laplacian dissipation, in supercritical Sobolev spaces, thereby revealing strong ill-posedness of the system at this regularity level. A distinctive feature of our approach is the introduction of a novel geometric construction, termed the ``Magnetic-solo ansatz'', through which, for the ideal MHD system, norm inflation occurs exclusively in the magnetic field $b$ in $H^s$ with $0<s<\frac{5}{2}$, while the $H^s$-norm of the velocity field $u$ remains uniformly bounded. This asymmetric behavior shows that supercritical ill-posedness can be driven exclusively by the magnetic field, highlighting its essential role in the breakdown of well-posedness. Our findings fill a significant gap in the supercritical regularity theory for incompressible MHD and shed light on the distinct mechanisms governing the fluid and magnetic dynamics.

math.AP↗

arXiv2026

Invariant Measure of the Camassa-Holm Equation with Linear Multiplicative Noise

In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise $$ \left\{ \begin{array}{l} {\rm d}u+(u\partial_xu+\partial_xP[u])\,{\rm d}t=βu\,{\rm d}W, u(0,x)=u_0(x), P[u]=(1-\partial_x^2)^{-1}\left(u^2+\frac 1 2(\partial_x u)^2\right) \end{array} \right. $$ depends almost surely continuously on the deterministic initial data in $H^s$ for $s>3/2$. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise.

math.AP↗

arXiv2026

Global Existence of Weak Martingale Solutions to the Camassa-Holm Equation with Linear Multiplicative Noise

In this paper, we consider the global existence and properties of $H^1$ martingale solution to the Camassa-Holm equation with linear multiplicative noise under periodic boundary conditions. The solution is obtained as limit of regular viscous approximate solutions to parabolic SPDEs, which are constructed using the Galerkin approximations ans the stochastic compactness method. The proof of convergence to a solution argues via tightness of the laws of the viscous approximations and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. In particular, by means of the Girsanov-type transform for regular viscous approximations and the convergence of Skorokhod-Jakubowski representations, we are able to establish the one-sided supernorm estimate and space-time higher regularity of the first-order spatial derivative, and large-time behavior of the weak martingale solution in the stochastic framework.

math.AP↗

arXiv2026

Non-uniqueness for the hyperdissipative Navier-Stokes equations with arbitrarily small subcritical data

In this paper, we consider the hyperdissipative Navier-Stokes equations with fractional dissipation $(-Δ)^β$ with $β>1$. We prove that smooth solutions of the hyperdissipative Navier-Stokes equations are non-unique with arbitrarily small initial data in ${B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ for any $α>0$. Moreover, we show the existence of a solution with arbitrarily small initial data in ${B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ ($α>0$) that grows arbitrarily large in $\dot{B}^{-s}_{\infty,\infty}(\mathbb{T}^d)$ for all $s\in\mathbb{R}$ in arbitrarily small time. It is worth pointing out that ${B}^{-β-α}_{\infty,1}(\mathbb{T}^d)$ lies in the subcritical regime when $0<α<β-1$. To the best of our knowledge, this is the first non-uniqueness result of the Navier-Stokes equations with initial data at the subcritical regularity. To show the sharpness of the above results, we establish the local well-posedness of the hyperdissipative Navier-Stokes equations with initial data in $\dot{B}^{-β-α}_{\infty,\infty}(\mathbb{T}^d)$ with $α< 0$.

math.AP↗

arXiv2026

Existence and uniqueness of the global conservative solutions for the generalized Camassa-Holm equation with dual-power nonlinearities

In this paper, we investigate the global conservative solutions to the generalized Camassa-Holm equation with dual-power nonlinearities. By introducing a new set of variables, we transform the original equation into an equivalent semi-linear system, which allows us to establish the global existence of conservative solutions. Furthermore, for a given global conservative solution, we construct some auxiliary variables tailored to its specific structure and demonstrate that they satisfy a semi-linear system with a unique solution, thereby deriving the uniqueness of conservative solutions to the original equation.

math.AP↗

arXiv2026

L2A: Learning to Accumulate Pose History for Accurate 3D Human Pose Estimation

Existing 2D-3D lifting human pose estimation methods have achieved strong performance. But the utilization of historical pose representations across network depth was overlooked. In current pipelines, information is propagated through fixed residual connections, which restricts effective reuse of early-layer features such as fine-grained spatial structures and short-term motion cues. However, naively incorporating historical features across layers is non-trivial. We further identify that maintaining a consistent representation space across layers is a prerequisite for effective cross-layer feature aggregation. To address this issue, we propose a history-aware framework that enables effective network cross-layer history feature utilization. Specifically, we adopt a spatial-temporal parallel Transformer backbone to prevent alternating spatial-temporal transformations during sequential processing, thereby maintaining a consistent representation space. Building upon this, we introduce a History Pose Accumulation (HPA) mechanism that adaptively aggregates features from all preceding layers to enhance current representations. Furthermore, we propose a Layer Pose History Aggregation (LPA) module that transforms layer pose features into a compact and structured form, reducing redundancy and enabling more stable aggregation. Extensive experiments demonstrate that our approach achieves state-of-the-art performance on benchmarks.

cs.CV↗

arXiv2026

Blow-up phemomenon for the 3-component Degasperis-Procesi equation

In this paper, we consider the Cauchy problem of the 3-component Degasperis-Procesi equation. Firstly, we discuss a local well-posedness result and a blow-up criterion in the low besov space. Secondly, we study the blow-up phenomenon by using the method which does not require any conservation law. Finally, we investigate some persistence properties.

math.AP↗

arXiv2026

Non-uniqueness of smooth solutions of the 5D magnetohydrodynamic equations from critical data

Recently, Coiculescu and Palasek \cite{Coiculescu2025} shows the non-uniqueness of solutions for the 3D incompressible Navier-Stokes equations with initial data in $BMO^{-1}$. Inspired by their breakthrough work, we develop their schemes for the incompressible magnetohydrodynamic equations and obtain a similar result in 5 dimensional case. More precisely, we construct two distinct global solutions with a initial data, which has nonvanishing velocity and magnetic fields in $BMO^{-1}(\mathbb{T}^5)$.

math.AP↗

arXiv2026

Global conservative weak solutions and global strong solutions for a class of weakly dissipative nonlinear dispersive wave equations

In this paper, we study the global existence of solutions of the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations $u_t-u_{xxt}+(f\left(u\right))_x-(f\left(u\right))_{xxx}+\left(g\left(u\right)+\frac{f^{\prime\prime}\left(u\right)}{2}u_x^2\right)_x+λ\left(u-u_{xx}\right)=0$. This includes the weakly dissipative Camassa-Holm equation and the weakly dissipative hyperelastic rod wave equation as special cases. Specifically, we establish three global existence results: one concerning the energy conservative weak solutions in a time-weighted $H^1$ space, and the other two concerning strong solutions, which include the cases of small initial data and sign-changing initial data. Our results recover and extend many known results for several classical models.

math.AP↗

arXiv2026

Global regularity and sharp decay rates to the 1D hypo-viscous compressible Navier-Stokes equations

In this paper, we study the global regularity and sharp decay rates for the isentropic hypo-viscous compressible Navier-Stokes equations in 1D. Firstly, we prove the global stability for the small initial data near a stable equilibrium. Especially, we establish the global critical regularity in the Sobolev space $H^β$ with $\frac{1}{2}<β<1$. Furthermore, by bootstrap argument, Fourier splitting method and energy method, we then establish the optimal time decay rates under the extra low-frequency smallness assumption. We find the $L^2$ energy is self-closed, which motivates us to obtain the existence of global large solutions for initial data with high regularity. By a pure energy method, we also derive the optimal time decay rates when $\frac{1}{2}\leβ<\frac{3}{4}$. We find a phenomenon that $\|(a,u)\|_{L^2}$ still decays even if the initial data does not possess $L^2$ smallness. Notably, the low-frequency smallness assumption is removed in the case with $\frac{1}{2}\leβ<\frac{3}{4}$.

math.AP↗

arXiv2026

Blow-up phemomenon for the Geng-Xue system and related models

In this paper, we consider the Cauchy problem of the Geng-Xue system with cubic nonlinearity. Firstly, we prove a blow-up criteria in the low besov space. Secondly, we prove the blow-up phenomenon by using the method which does not require any conservation law. Finally, we extend our results to the b-family of two-component system with cubic nonlinearity.

math.AP↗

arXiv2026

Global regularity and sharp decay to the 2D Hypo-Viscous compressible Navier-Stokes equations

In this paper, we consider the global regularity and the optimal time decay rate for the 2D isentropic hypo-viscous compressible Navier-Stokes equations. Firstly, we prove that there exists a global strong solution with the small initial data are close to the constant equilibrium state in $H^s$ framework with $s>1$. Furthermore, by virtue of improved Fourier splitting method and the Littlewood-Paley decomposition theory, we then establish the optimal time decay rate for low regularity data.

math.AP↗

arXiv2025

Uniform vanishing damping limit for the 2D inviscid Oldroyd-B model with fractional stress tensor diffusion

This paper is devoted to the uniform vanishing damping limit of the 2D inviscid Oldroyd-B model with fractional stress tensor diffusion. Firstly, we find that fractional stress tensor diffusion helps to reduce the global regularity of the 2D Oldroyd-B model with damping coefficient $a\in[0,1]$. By virtue of improved Fourier splitting method, we then prove the optimal time decay rates under the critical regularity for $a=0$. When $a\in (0,1]$, we establish time decay rates that are uniform with respect to $a$. Combining the time decay rate for $a\in [0,1]$ and the time integrability, we obtain the uniform damping vanishing rates for the 2D Oldroyd-B model. Using spectral analysis methods, we finally improve the time decay rates for $\mathrm{tr}τ$ with $a\in (0,1]$, which ensure the sharp uniform damping vanishing rates of $\mathrm{tr}τ$.

math.AP↗

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.

math.AP↗

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 $L^{\frac{2α}{2α-1}}_TL^\infty_x$.

math.AP↗

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.

math.AP↗