Searcharxiv⌕ Search

arXiv subjects

Daoguo Zhou

Publications and source records attributed to Daoguo Zhou.

10 recordsLinked to original sources
arXiv2023

Remarks on Interior Regularity Criteria Without Pressure for the Navier-Stokes Equations

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the $L^p_tL_x^q(Q_1)$ norm of the velocity is sufficiently small, where $1\leq \frac{2}{p}+\frac{3}{q}<2$ and $2\leq p\leq \infty$. It improves the recent result of $p,q>2 $ by Kwon \cite{Kwon} (J. Differential Equations 357 (2023), 1--31.), and also generalizes Chae-Wolf's $L_t^\infty L_x^{\frac32+}$ criterion \cite{CW2017} (Arch. Ration. Mech. Anal. 225 (2017), no. 1, 549--572.).

math.AP↗
arXiv2022

Partial regularity of suitable weak solutions of the model arising in amorphous molecular beam epitaxy

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $\alpha$ in the nonlinear term in the following parabolic equation $$h_t+h_{xxxx}+\partial_{xx}|h_x|^\alpha=f.$$ It is shown that when $5/3\leq\alpha<7/3$, the $\frac{3\alpha-5}{\alpha-1}$-dimensional parabolic Hausdorff measure of $\mathcal{S}$ is zero, which generalizes the recent corresponding work of Oz\'anski and Robinson in [31,SIAM J. Math. Anal. 51: 228--255, 2019] for $\alpha=2$ and $f=0$. The same result is valid for a 3D modified Navier-Stokes system.

math.AP↗
arXiv2021

On Endpoint Regularity Criterion of the 3D Navier-Stokes equations

Let $(u, π)$ with $u=(u_1,u_2,u_3)$ be a suitable weak solution of the three dimensional Navier-Stokes equations in $\mathbb{R}^3\times [0, T]$. Denote by $\dot{\mathcal{B}}^{-1}_{\infty,\infty}$ the closure of $C_0^\infty$ in $\dot{B}^{-1}_{\infty,\infty}$. We prove that if $u\in L^\infty(0, T; \dot{B}^{-1}_{\infty,\infty})$, $u(x, T)\in \dot{\mathcal{B}}^{-1}_{\infty,\infty})$, and $u_3\in L^\infty(0, T; L^{3, \infty})$ or $u_3\in L^\infty(0, T; \dot{B}^{-1+3/p}_{p, q})$ with $3<p, q< \infty$, then $u$ is smooth in $\mathbb{R}^3\times [0, T]$. Our result improves a previous result established by Wang and Zhang [Sci. China Math. 60, 637-650 (2017)].

math.AP↗
arXiv2020

Regularity of Solutions to the Navier-Stokes equations in $\dot{B}_{\infty,\infty}^{-1}$

We prove that if $u$ is a suitable weak solution to the three dimensional Navier-Stokes equations from the space $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, then all scaled energy quantities of $u$ are bounded. As a consequence, it is shown that any axially symmetric suitable weak solution $u$, belonging to $L_{\infty}(0,T;\dot{B}_{\infty,\infty}^{-1})$, is smooth.

math.AP↗
arXiv2019

New $\varepsilon$-regularity criteria of suitable weak solutions of the 3D Navier-Stokes equations at one scale

In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new $\varepsilon$-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale: for any $p,q\in [1,\infty]$ satisfying $1\leq 2/q+3/p <2$, there exists an absolute positive constant $\varepsilon$ such that $u\in L^{\infty}(Q(1/2))$ if $$\|u\|_{L^{p,q}(Q(1))}+\|Π\|_{L^{1 }(Q(1))}<\varepsilon.$$ This is an improvement of corresponding results recently proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017]. As an application of these $\varepsilon$-regularity criteria, we improve the known upper box dimension of the possible interior singular set of suitable weak solutions of the Navier-Stokes system from $975/758(\approx1.286)$ [28] to $2400/1903 (\approx1.261)$.

math.AP↗
arXiv2019

$\varepsilon$-regularity criteria in anisotropic Lebesgue spaces and Leray's self-similar solutions to the 3D Navier-Stokes equations

In this paper, we establish some $\varepsilon$-regularity criteria in anisotropic Lebesgue spaces for suitable weak solutions to the 3D Navier-Stokes equations as follows: $$ \limsup\limits_{\varrho\rightarrow0} \varrho^{1-\frac{2}{p}-\sum\limits^{3}_{j=1}\frac{1}{q_{j}}} \|u\|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(\varrho))} \leq\varepsilon, ~~\frac{2}{p}+\sum\limits^{3}_{j=1}\frac{1}{q_{j}} \leq2~~~~~\text{with}~q_{j} > 1;\\$$$$ \sup_{-1\leq t\leq0}\|u\|_{L^{\overrightarrow{q}}(B(1))} < \varepsilon,~~\frac{1}{q_{1}}+\frac{1}{q_{2}}+\frac{1}{q_{3}} <2\quad \text{with}\, 1<q_{j}<\infty;$$ $$\|u \|_{L_{t}^{p}L^{\overrightarrow{q}}_{x}(Q(1))} +\|Π\|_{L^{1}(Q(1))}\leq\varepsilon, \quad \frac2p+\sum^{3}_{j=1}\frac{1}{q_{j}} <2 ~~~\text{with}~~ 1<q_{j}<\infty, $$ which extends the previous results in [2, 12, 18, 19, 22, 37, 43]. As an application, in the spirit of [4], we prove that there does not exist a nontrivial Leray's backward self-similar solution with profiles in $L^{\overrightarrow{p}}(\mathbb{R}^{3})$ with $\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}<2$. This generalizes the corresponding results of [4, 20, 28, 38].

math.AP↗
arXiv2018

A $\varepsilon$-regularity criterion without pressure of suitable weak solutions to the Navier-Stokes equations at one scale

In this paper, we continue our work in [15] to derive ε-regularity criteria at one scale without pressure for suitable weak solutions to the Navier-Stokes equations. We establish a $\varepsilon$-regularity criterion below of suitable weak solutions, for any $δ>0$, $$\iint_{Q(1)}|u|^{\frac{5}{2}+δ}dxdt\leq \varepsilon.$$ As an application, we extend the previous corresponding results concerning the improvement of the classical Caffarelli--Kohn--Nirenberg theorem by a logarithmic factor.

math.AP↗
arXiv2018

On Wolf's regularity criterion of suitable weak solutions to the Navier-Stokes equations

In this paper, we consider the local regularity of suitable weak solutions to the 3D incompressible Navier-Stokes equations. By means of the local pressure projection introduced by Wolf in [15,16], we present a $\varepsilon$-regularity criterion below of suitable weak solutions $$ \iint_{Q(1)}|u|^{20/7}dxdt\leq \varepsilon, $$ which gives an improvement of previous corresponding results obtained in Chae and Wolf [3, Arch. Ration. Mech. Anal., 225: 549-572, 2017], in Guevara and Phuc [6, Calc. Var., 56:68, 2017] and in Wolf [16, Ann. Univ. Ferrara, 61: 149-171, 2015].

math.AP↗
arXiv2016

Global well-posedness for the 2D Boussinesq Equations with Zero Viscosity

We prove the global well-posedness of the two-dimensional Boussinesq equations with zero viscosity and positive diffusivity in bounded domains for rough initial data [ $u_{0}\in L^{2}$, $\text{curl}\,u_{0}\in L^{\infty}$ and $θ_{0}\in B^{2-2/p}_{q,p}$ with $p\in (1,\infty)$, $q\in (2,\infty)$ ]. Our method is based on the maximal regularity for heat equation.

math.AP↗