arXiv subjects

Zhuan Ye

Publications and source records attributed to Zhuan Ye.

At least 19 recordsLinked to original sources

arXiv2026

Global regularity of the 2D fractional Boussinesq equations with subcritical dissipation

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by $(-Δ)^{\frac\alpha2}u$ and $(-Δ)^{\frac\beta2} θ$. Attention is focused on the subcritical regime where $α+ β>1$. The case $α>\frac23$ was recently settled in a joint work of the authors [Math. Ann., \textbf{391} (2025), 5965-6012], which established global regularity under this condition. This paper addresses the remaining case $α\leq \frac23$. We obtain the sharpest regularity result by minimizing assumptions on $α$ and $β$. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.

math.AP↗

arXiv2025

Global well-posedness of the 2D primitive equations with fractional horizontal dissipation

In this paper, we investigate the two-dimensional incompressible primitive equations with fractional horizontal dissipation. Specifically, we establish global well-posedness of strong solutions for arbitrarily large initial data when the dissipation exponent satisfies $α\geqα_{0}\approx1.1108$. In addition, we prove global well-posedness of strong solutions for small initial data when $α\in [1, α_0)$. Notably, the smallness assumption is imposed only on the $L^\infty$ norm of the initial vorticity.

math.AP↗

arXiv2025

On the Fermat-type partial differential-difference equations on $\mathbb{C}^n$

Assume that $n$ is a positive integer, $p_{j}$ ($j=1,2, \cdots, 6)$ are polynomials, $p$ is an irreducible polynomial, and $f$ is an entire function on $\mathbb{C}^{n}.$ Let $ L(f)=\sum_{j=1}^s q_{t_j}f_{z_{t_j}}$ and $\overline{f}(z)=f(z_{1}+c_{1}, \ldots, z_{n}+c_{n})$, where $q_{t_j}$ ($j=1,2, \cdots, s\le n$) are non-zero polynomials on $\mathbb{C}^{n}$ and $c=(c_{1}, \ldots, c_{n})\in \mathbb{C}^{n}\setminus\{0\}$. We show the structures of all entire solutions to the non-linear partial differential-difference equation $$(p_{1} L(f)+p_{2}\overline{f}+p_5 f)^{2}+(p_{3}L(f)+p_{4}\overline{f}+p_6 f)^{2}=p.$$ The partial differential-difference equation is called a Fermat-type partial differential-difference equation (PDDE). Further, we find many sufficient conditions and/or necessary conditions for the existence, as well as the concrete representations, of entire solutions to the Fermat-type PDDE. We also demonstrate several examples on $\mathbb{C}^2$ with non-constant coefficients to verify that all representations in our theorems exist and are accurate and that the entire solutions to the Fermat-type PDDEs could have finite or infinite growth order. Our theorems unify and extend previous results (see, e.g., [2, 3, 10, 12, 32]).

math.CV↗

arXiv2023

Global well-posedness for the 2D Euler-Boussinesq-B$\rm\acute{e}$nard equations with critical dissipation

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-B$\rm\acute{e}$nard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (J. Differential Equations 253:100--125, 2012).

math.AP↗

arXiv2023

Remark on the global regularity of 2D MHD equations with almost Laplacian magnetic diffusion

Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion (without velocity dissipation) are globally well-posed is a difficult problem and remains completely open. In this paper, we establish the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regularity issue on the 2D MHD equations with standard Laplacian magnetic diffusion.

math.AP↗

arXiv2023

Global regularity of 2D generalized incompressible magnetohydrodynamic equations

In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-Δ)^α$ and magnetic diffusion given by reducing about logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power $α$ is a positive constant. In addition, we prove several global \emph{a priori} bounds for the case $α=0$. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global regularity issue on the 2D resistive MHD equations, namely, the case when the MHD equations only have standard Laplacian magnetic diffusion.

math.AP↗

arXiv2021

Global regularity and time decay for the SQG equation with anisotropic fractional dissipation

In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. On the one hand, when the dissipation powers are restricted to a suitable range, the global regularity of the surface quasi-geostrophic equation is obtained by some anisotropic embedding and interpolation inequalities involving fractional derivatives. One the one hand, we obtain the optimal large time decay estimates for global weak solutions by an anisotropic interpolation inequality. Moreover, based on the argument adopted in establishing the global $\dot{H}^1$-norm of the solution, we obtain the optimal large time decay estimates for the above obtained global smooth solutions. Finally, the decay estimates for the difference between the full solution and the solution to the corresponding linear part are also derived.

math.AP↗

arXiv2020

Inequalities Concerning Maximum Modulus and Zeros of Random Entire Functions

Let $f_ω(z)=\sum\limits_{j=0}^{\infty}χ_j(ω) a_j z^j$ be a random entire function, where $χ_j(ω)$ are independent and identically distributed random variables defined on a probability space $(Ω, \mathcal{F}, μ)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Steinhaus entire functions. Then, we prove that, for almost all functions in the family and for any constant $C>1$, there exist a constant $r_0=r_0(ω)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r>r_0$ and $r\notin E$, $$ |\log M(r, f)- N(r,0, f_ω)|\le (C/A)^{\frac1{B}}\log^{\frac1{B}}\log M(r,f) +\log\log M(r, f), \qquad a.s. $$ where $A, B$ are constants, $M(r, f)$ is the maximum modulus, and $N(r, 0, f)$ is the weighted counting-zero function of $f$. As a by-product of our main results, we prove Nevanlinna's second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by a weighed counting function, rather than by two weighted counting functions in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions $f_ω$ and for any $ε>0$, there is $r_0$ such that, for $r>r_0$, $$ T(r, f) \le N(r,0, f_ω)+(\frac12+ε) \log T(r, f). $$

math.CV↗

arXiv2020

Global regularity of the three-dimensional fractional micropolar equations

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Δ)^αu$ and $(-Δ)^βw$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $α\geq \frac{5}{4}$, $β\geq 0$ and $α+β\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.

math.AP↗

arXiv2019

On the global regularity for anisotropic dissipative surface quasi-geostrophic equation

In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. Global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply, to overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the classical Gronwall inequality, we establish a new logarithmic type Gronwall inequality, which may be of independent interest and potential applications.

math.AP↗

arXiv2019

Global regularity of the two-dimensional regularized MHD equations

In this paper, we consider the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations. The main objective of this paper is to establish the global regularity of classical solutions of the magnetohydrodynamics equations with the minimal dissipation. Consequently, our results significantly improve the previous works.

math.AP↗

arXiv2019

Global regularity of 2D tropical climate model with zero thermal diffusion

This article studies the global regularity problem of the two-dimensional zero thermal diffusion tropical climate model with fractional dissipation, given by $(-Δ)^αu$ in the barotropic mode equation and by $(-Δ)^βv$ in the first baroclinic mode of the vector velocity equation. More precisely, we show that the global regularity result holds true as long as $α+β\geq2$ with $1<α<2$. In addition, with no dissipation from both the temperature and the first baroclinic mode of the vector velocity, we also establish the global regularity result with the dissipation strength at the logarithmically supercritical level. Finally, our arguments can be extended to obtain the corresponding global regularity results of the higher dimensional cases.

math.AP↗

arXiv2018

Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier-Stokes equations with vacuum

The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.

math.AP↗

arXiv2016

On the Differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion

We investigate the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms of the spatial Hölder regularity, we prove that the vanishing viscosity solution is differentiable with some Hölder continuous derivatives for any positive time.

math.AP↗

arXiv2015

Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation

As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [12]. Consequently, this result significantly improves the recent works [12, 38, 40].

math.AP↗

arXiv2015

Global smooth solution to the 2D Boussinesq equations with fractional dissipation

In this paper, we consider the two-dimensional (2D) incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. Based on the previous works and some new observations, we show that the condition $1-α<β<\min\Big\{3-3α,\,\,\fracα{2},\,\, \frac{3α^{2}+4α-4}{8(1-α)}\Big\}$ with $0.7351\approx\frac{10-2\sqrt{10}}{5}<α<1$ suffices in order for the solution pair of velocity and temperature to remain smooth for all time.

math.AP↗

arXiv2015

Global regularity for the 2D Oldroyd-B model in the corotational case

This paper is dedicated to the Oldroyd-B model with fractional dissipation $(-Δ)^ατ$ for any $α>0$. We establish the global smooth solutions to the Oldroyd-B model in the corotational case with arbitrarily small fractional powers of the Laplacian in two spatial dimensions. The methods described here are quite different from the tedious iterative approach used in recent paper \cite{XY}. Moreover, in the Appendix we provide some a priori estimates to the Oldroyd-B model in the critical case which may be useful and of interest for future improvement. Finally, the global regularity to to the Oldroyd-B model in the corotational case with $-Δu$ replaced by $(-Δ)^γu$ for $γ>1$ are also collected in the Appendix. Therefore our result is more closer to the resolution of the well-known global regularity issue on the critical 2D Oldroyd-B model.

math.AP↗

arXiv2015

A note on global regularity results for 2D Boussinesq equations with fractional dissipation

In this paper we study the Cauchy problem for the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Invoking the energy method and several commutator estimates, we get the global regularity result of the 2D Boussinesq equations as long as $1-α<β< \min\Big\{\fracα{2},\,\, \frac{3α-2}{2α^{2}-6α+5}, \,\,\frac{2-2α}{4α-3}\Big\}$ with $0.77963\thickapproxα_{0}<α<1$. As a result, this result is a further improvement of the previous two works \cite{MX,YXX}.

math.AP↗