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Liqun Zhang

Publications and source records attributed to Liqun Zhang.

At least 19 recordsLinked to original sources
arXiv2026

On the Regularity of Navier-Stokes Equations in Critical Space

This paper focuses on the regularity of the Navier-Stokes equations in critical space. Let $ u(x,t) $ and $ p(x,t) $ denote suitable weak solution of the Navier-Stokes equations in $Q_T=\mathbb{R}^3\times(-T, 0)$. We prove that if $u(x,t)$ is in the scaling invariant spaces $L_t^{\infty}L_{x_3}^{p_1}L_{x_h}^{p_2}(Q_T)$ , where $ \frac{1}{p_1}+\frac{2}{p_2}=1 $ , $p_1\geq 2$ and $ x_h = (x_1, x_2) $ , then $ u $ is a smooth solution in $ Q_T $ and doesn't blow up at $ t = 0 $. In particular, if $ u(x,t) \in L_t^{\infty}L_{x_3}^{\infty}L_{x_h}^{2}(Q_T)$, then $u(x,t)$ is a smooth solution in $ Q_T $ and regular up to $ t = 0 $.

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arXiv2023

Symmetric Stationary Boundary Layer

Considering the boundary layer problem in the case of two-dimensional flow past a wedge with the wedge angle $φ=π\frac{2m}{m+1}$, Oleinik and Samokhin obtained the local well-posedness results for $m \geq 1$. In this paper, we establish the existence and uniqueness of classical solutions to the Prandtl systems for arbitrary $m>0$, which solves the steady case in Open problem 6 proposed by Oleinik and Samokhin. Our proof is based on the maximum principle technique at the Crocco coordinates and the most important observation that when the fluid approaches a sharp point, it seems the self-similar solutions. Then we obtain the existence and uniqueness of the solution with the help of the self-similar solutions by the Line Method. Furthermore, we similarly establish the well-posedness results of three-dimensional flow past a cone.

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arXiv2022

Existence of blowup solutions to Boussinesq equations on $\mathbb{R}^3$ with dissipative temperature

The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion. Recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,α}$ finite time blow-up solutions for Euler systems with finite energy. Inspired by their works, we constructed $C^{1,α}$ finite time blow-up solution for Boussinesq equations where the temperature has diffusion and finite energy. Generally speaking, the diffusion of temperature smooths the solution of the system which is against the formations of singularity. The main difficulty is that the Laplace operator of the temperature equation is not coercive under the Sobolev weighted norm introduced by Elgindi. We introduced a new time depending scaling formulation and new weighted Sobolev norms, under which we obtain the nonlinear estimate. The new norm is well-coupled with the original norm, which enables us to finish the proof.

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arXiv2022

Global Well-posedness and Regularity of Weak Solutions to the Prandtl's System

We continue our study on the global solution to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik provided that the pressure is favorable. First, by using a different method from [13], we gave a direct proof of existence of a global weak solution by a direct BV estimate. Then we prove the uniqueness and continuous dependence on data of such a weak solution to the initial-boundary value problem. Finally, we show the smoothness of the weak solutions and then the global existence of smooth solutions.

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arXiv2021

Remarks on the Steady Prandtl Boundary Layer Expansions

We continue the study of the validity of the Prandtl boundary layer expansions in \cite{GZ}, where by estimating the stream-function of the remainder, we proved if the Euler flow is perturbation of shear flows when the width of domain is small. In this paper, we obtain a new derivatives estimate of stream-function away from the boundary layer and then prove the validity of expansions for any non-shear Euler flow, provided the width of domain is small.

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arXiv2020

Resonant Decompositions and Global Well-posedness for 2D Zakharov-Kuznetsov Equation in Sobolev spaces of Negative Indices

The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown to be global well-posed for the initial date in $H^{s}$ provided $s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below $L^2$, we construct an almost conserved quantity using multilinear correction term following the $I$-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao. In contrast to KdV equation, the main difficulty is to handle the resonant interactions which are significant due to the multidimensional and multilinear setting of the problem. The proof relies upon the bilinear Strichartz estimate and the nonlinear Loomis-Whitney inequality.

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arXiv2020

On the Steady Prandtl Boundary Layer Expansions

In this paper, we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in $(0,L)\times\mathbb{R}^+$ with non-slip boundary conditions. By estimating the stream-function of the remainder, we justify the validity of the Prandtl boundary layer expansions.

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arXiv2019

$C^α$ regularity of weak solutions of non-homogenous ultraparabolic equations with drift terms

Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are Hölder continuous, which also generalizes the classic results of parabolic equations of second order. The main ingredients are a type of weak Poincaré inequality satisfied by non-negative weak sub-solutions and Moser iteration.

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arXiv2019

Hölder continuous weak solution of 2d Boussinesq equation with diffusive temperature

We show the existence of Hölder continuous periodic weak solutions of the 2d Boussinesq equation with diffusive temperature which satisfy the prescribed kinetic energy. More precisely, for any smooth $e(t):[0,1]\rightarrow R_+$ and $\varepsilon\in (0, \frac{1}{10})$, there exist $v\in C^{\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2), θ\in C_t^{1,\frac{1}{20}-\frac{\varepsilon}{2}}C_x^{2,\frac{1}{10}-\varepsilon}([0,1]\times {\rm T}^2)$ which solve boussinesq equation in the sense of distribution and satisfy e(t)=\int_{{\rm T}^2}|v(t,x)|^2dx, \quad \forall t\in [0,1].

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arXiv2017

Backward uniqueness for general parabolic operators in the whole space

We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. This result extends in some ways a classical result of Lions and Malgrange [12] and a recent result of the authors [10].

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arXiv2017

Backward uniqueness for parabolic operators with variable coefficients in a half space

It is shown that a function $u$ satisfying $|\partial_tu+\sum_{i,j}\partial_i(a^{ij}\partial_ju)|\leq N(|u|+|\nabla u|)$, $|u(x,t)|\leq Ne^{N|x|^2}$ in $\mathbb{R}^n_+\times[0,T]$ and $u(x,0)=0$ in $\mathbb{R}^n_+$ under certain conditions on $\{a^{ij}\}$ must vanish identically in $\mathbb{R}^n_+\times[0,T]$. The main point of the result is that the conditions imposed on $\{a^{ij}\}$ are of the type: $\{a^{ij}\}$ are Lipschitz and $|\nabla_xa^{ij}(x,t)|\leq \frac{E}{|x|}$, where $E$ is less than a given number, and the conditions are in some sense optimal.

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arXiv2015

On The Continuous Periodic Weak Slution of Boussinesq Equations

The Boussingesq equations was introduced in understanding the coupling nature of the thermodynamics and the fluid dynamics. We show the existence of continuous periodic weak solutions of the Boussinesq equations which satisfies the prescribed kinetic energy or some other prescribed property. Our results represent the conversions between internal energy and mechanical energy.

math.AP↗
arXiv2015

Landis-Oleinik Conjecture in the Exterior Domain

In 1974, Landis and Oleinik conjectured that if a bounded solution of a parabolic equation decays fast at a time, then the solution must vanish identically before that time, provided the coefficients of the equation satisfy appropriate conditions at infinity. We prove this conjecture under some reasonable assumptions on the coefficients which improved the earlier results.

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