arXiv subjects

Zhaonan Luo

Publications and source records attributed to Zhaonan Luo.

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.

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arXiv2026

Non-implosion mechanism of 3D incompessible Euler equations

This paper studies the non-implosion mechanism for the 3D incompressible Euler equations. We prove that vorticity blows up in finite time, whereas the $L^p_T L^\infty_{loc}$ $(p\in[1,\infty))$ norm of the velocity field remains bounded. Moreover, under an appropriate assumption on the scaling index, the exponent $p$ can be taken to be infinite. The proof is based on the introduction of a refined framework, the new observations for the null structure of transport term, and stability analysis of the self-similar model.

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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

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.

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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 $\mathrm{tr}τ$.

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arXiv2025

A Besov-based integration-by-parts method for the incompressible Navier-Stokes equations

This note introduces a novel numerical analysis framework for the incompressible Navier-Stokes equations based on Besov spaces. The key contribution of this note is to establish the stability and convergence of a semi-implicit time-stepping scheme by deriving precise error estimates in the $B^0_{\infty,1}$ and $B^0_{\infty,2}$ spaces. Another contribution of our analysis is the detailed treatment of the $B^0_{\infty,2}$ case, where a crucial integration-by-parts technique is employed to adeptly handle the nonlinear advection term. This technique allows for a refined estimate that effectively transfers derivatives onto the test functions, mitigating the inherent analytical challenges posed by the low regularity of these spaces. Our results provide sharper, more localized error bounds than in classical Sobolev spaces, directly linking the scheme's convergence to the critical regularity of the continuous solution. This work underscores the advantage of Besov spaces for the numerical analysis of nonlinear fluid PDEs.

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arXiv2024

Global well-posedness and uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model

In this paper, we consider global strong solutions and uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model in R^d, where d=2 and 3. The well-recognized problem of the global existence of smooth solutions for the 2D inviscid Oldroyd-B model without smallness assumptions is open due to the complex structure of Q. Therefore improving the smallness assumptions, especially in lower regularity class, is the core question in the area of fluid models. On the other hand, long-time behaviors of solutions including temporal decay and uniform-in-time damping stability are also of deep significance. These problems have been widely studied, however, the existing results are not regularity critical and the (uniform) vanishing damping limit has not been discussed. The goal of this work is to dig deeper in this direction. In this work we first establish the local well-posedness in the sense of Hadamard with critical regularity. Then, by virtue of the sharp commutator estimate for Calderon-Zygmund operator, we establish the global existence of solutions for d=2 with damping in the low regularity class, which to our best knowledge, is novel in the literature. Furthermore, in both 2D and 3D cases, we prove the global existence of the solutions to the inviscid Oldroyd-B model independent of the damping parameters. In addition, we obtain the optimal temporal decay rates and time integrability by improving the existing Fourier splitting method and developing a novel decomposition strategy. One of the major contributions of the presenting paper is to prove the uniform-in-time vanishing damping limit for the inviscid Oldroyd-B model and discover the correlation between sharp vanishing damping rate and the temporal decay rate. Finally, we will support our findings by providing numerical evidence regarding the vanishing damping limit in the periodic domain T^d.

math.AP↗

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.

math.AP↗

arXiv2024

On semi-implicit schemes for the incompressible Euler equations via the vanishing viscosity limit

A new type of systematic approach to study the incompressible Euler equations numerically via the vanishing viscosity limit is proposed in this work. We show the new strategy is unconditionally stable that the $L^2$-energy dissipates and $H^s$-norm is uniformly bounded in time without any restriction on the time step. Moreover, first-order convergence of the proposed method is established including both low regularity and high regularity error estimates. The proposed method is extended to full discretization with a newly developed iterative Fourier spectral scheme. Another main contributions of this work is to propose a new integration by parts technique to lower the regularity requirement from $H^4$ to $H^3$ in order to perform the $L^2$-error estimate. To our best knowledge, this is one of the very first work to study incompressible Euler equations by designing stable numerical schemes via the inviscid limit with rigorous analysis. Furthermore, we will present both low and high regularity errors from numerical experiments and demonstrate the dynamics in several benchmark examples.

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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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

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.

math.AP↗

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 $d\geq 3$.

math.AP↗