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Baishun Lai

Publications and source records attributed to Baishun Lai.

At least 19 recordsLinked to original sources
arXiv2026

Forward self-similar solutions of the Navier-Stokes equations in R^n

We investigate forward self-similar solutions to the incompressible Navier--Stokes equations with homogeneous initial data $u_0(x)=σ(x/|x|)/|x|$, $x\in\mathbb R^n$, focusing on the critical four-dimensional case. For the existence theory in dimensions $3\leq n\leq5$, we develop a Galerkin scheme for the Leray profile problem, featuring a decomposition of the self-similar data and nonlocal heat kernel effects across spatial scales. Exploiting the structural properties of the data and the coercivity of the Leray operator, we derive uniform \textit{a priori} estimates, pass to the limit, and obtain weak forward self-similar solutions. In the 4D case, we prove new Stokes-type estimates for the linear Leray system. The inverse Leray operator admits an integral representation as a singular integral operator; its $L^p$-boundedness follows from Calderón--Zygmund theory. Combined with a 4D compactness estimate for the convective term, this closes the regularity iteration. For spatial decay, we reformulate the problem in weighted spaces. The non-integrability of the low-frequency kernel causes a logarithmic loss under $σ\in W^{1,\infty}(\mathbb S^3)$. With $σ\in C^{1,γ}(\mathbb S^3)$, $0<γ<1$, frequency-localized estimates remove this loss and yield sharp decay.

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arXiv2026

Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values

This paper is concerned with the weak solution theory for the MHD system with large $L^3$-initial data. Due to the fact that the natural boundary condition on the magnetic field $H$ is the slip boundary condition, the Leray-Schauder fixed-point theorem, which have used to investigate the weak solution theory of the Navier-Stokes system, becomes invalid. To address such difficulty, we will invoke the Leray's approximation technique and the perturbation theory to seek a global weak solution to the Cauchy problem for MHD equations with large $L^3$-initial data. Our strategy provides a simple alternative (self-contained) proof of weak $L^3$-solution theory of incompressible Navier-Stokes system. Moreover, this weak solution is unique under some restrictions.

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arXiv2024

Quantitative regularity for the MHD equations via the localization technique in frequency space

In this paper, we employ the localization technique in frequency space developed by Tao in \cite{MR4337421} to investigate the quantitative estimates for the MHD equations. With the help of quantitative Carleman inequalities given by Tao in \cite{MR4337421} and the pigeonhole principle, we establish the quantitative regularity for the critical $L^3$ norm bounded solutions which enables us explicitly quantify the blow-up behavior in terms of $L^3$ norm near a potential first-time singularity. Some technical innovations, such as introducing the corrector function, are required due to the fact that the scales are inconsistent between the magnetic field and the vorticity field.

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arXiv2024

Applications of the Green tensor estimates of the nonstationary Stokes system in the half space

In this paper, we present a series of applications of the pointwise estimates of the (unrestricted) Green tensor of the nonstationary Stokes system in the half space, established in our previous work [CMP 2023]. First, we show the $L^1$-$L^q$ estimates for the Stokes flow with possibly non-solenoidal $L^1$ initial data, generalizing the results of Giga-Matsui-Shimizu [Math. Z. 1999] and Desch-Hieber-Prüss [J. Evol. Equ. 2001]. Second, we construct mild solutions of the Navier-Stokes equations in the half space with mixed-type pointwise decay or with pointwise decay alongside boundary vanishing. Finally, we explore various coupled fluid systems in the half space including viscous resistive magnetohydrodynamics equations, a coupled system for the flow and the magnetic field of MHD type, and the nematic liquid crystal flow. For each of these systems, we construct mild solutions in $L^q$, pointwise decay, and uniformly local $L^q$ spaces.

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arXiv2022

The Green tensor of the nonstationary Stokes system in the half space

We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier-Stokes equations in $L^q$, pointwise decay, and uniformly local $L^q$ spaces in the half-space.

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arXiv2021

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

For the Stokes system in the half space, Kang [Math.~Ann.~2005] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in \cite{Green} (arXiv:2011.00134). We also examine the Stokes flows generated by dipole bumps boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

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arXiv2021

Global Regularity of weak solutions to the generalized Leray equations and its applications

We investigate a regularity for weak solutions of the following generalized Leray equations \begin{equation*} (-Δ)^αV- \frac{2α-1}{2α}V+V\cdot\nabla V-\frac{1}{2α}x\cdot \nabla V+\nabla P=0, \end{equation*} which arises from the study of self-similar solutions to the generalized Naiver-Stokes equations in $\mathbb R^3$. Firstly, by making use of the vanishing viscosity and developing non-local effects of the fractional diffusion operator, we prove uniform estimates for weak solutions $V$ in the weighted Hilbert space $H^α_ω(\mathbb R^3)$. Via the differences characterization of Besov spaces and the bootstrap argument, we improve the regularity for weak solution from $H^α_ω(\mathbb R^3)$ to $H_ω^{1+α}(\mathbb R^3)$. This regularity result, together linear theory for the non-local Stokes system, lead to pointwise estimates of $V$ which allow us to obtain a natural pointwise property of the self-similar solution constructed in \cite{LXZ}. In particular, we obtain an optimal decay estimate of the self-similar solution to the classical Naiver-Stokes equations by means of the special structure of Oseen tensor. This answers the question proposed by Tsai \cite[Comm. Math. Phys., 328 (2014), 29-44]{T}.

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arXiv2019

Forward self-similar solutions of the fractional Navier-Stokes Equations

We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion $(-Δ)^α.$ First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with $5/6<α\leq1$ for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this solution is smooth in $\mathbb R^3\times (0,+\infty)$. In particular, when $α=1$, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE 9 (2016), 1811-1827] satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in [Jia and Šverák, Invent. Math. 196 (2014), 233-265].

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arXiv2017

Well-posedness of a fourth order evolution equation Modeling MEMS

We consider a fourth order evolution equation involving a singular nonlinear term $\fracλ{(1-u)^{2}}$ in a bounded domain $Ω\subset\R^{n}$. This equation arises in the modeling of microelectromechanical systems. We first investigate the well-posedness of a fourth order parabolic equation which has been studied in \cite{Lau}, where the authors, by the semigroup argument, obtained the well-posedness of this equation for $n\leq2$. Instead of semigroup method, we use the Faedo-Galerkin technique to construct a unique solution of the fourth order parabolic equation for $n\leq7$, which improves and completes the result of \cite{Lau}. Besides, the well-posedness of the corresponding fourth order hyperbolic equation is obtained by the similar argument for $n\leq7$.

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arXiv2014

Remarks on two fourth order elliptic problems in whole space

We are interested in entire solutions for the semilinear biharmonic equation $Δ^{2}u=f(u)$ in $\R^N$, where $f(u)=e^{u}$ or $-u^{-p}\ (p>0)$. For the exponential case, we prove that any classical entire solution verifies $-Δu>0$ without any restriction, which completes the results in \cite{Dupaigne, xu-wei} and yields a nonexistence result in $\R^2$ ; we obtain also a refined asymptotic expansion of radial separatrix solution for $N=3$, which answers a question in \cite{Berchio}. For the negative power case, we show the nonexistence of the classical entire solution for any $0<p\leq1$.

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arXiv2013

Global Well-Posedness of the Landau-Lifshitz-Gilbert equation for initial data in Morrey space

We establish the global well-posedness of the Landau-Lifshitz-Gilbert equation in $\mathbb R^n$ for any initial data ${\bf m}_0\in H^1_*(\mathbb R^n,\mathbb S^2)$ whose gradient belongs to the Morrey space $M^{2,2}(\mathbb R^n)$ with small norm $\displaystyle\|\nabla {\bf m}_0\|_{M^{2,2}(\mathbb R^n)}$. The method is based on priori estimates of a dissipative Schrödinger equation of Ginzburg-Landau types obtained from the Landau-Lifshitz-Gilbert equation by the moving frame technique.

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arXiv2011

Properties of the extremal solution for a fourth-order elliptic problem

Let $λ^{*}>0$ denote the largest possible value of $λ$ such that $$ \{{array}{lllllll} Δ^{2}u=\fracλ{(1-u)^{p}} & \{in}\ \ B, 0 1$ and $n$ is the exterior unit normal vector. We show that for $λ=λ^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough and $1-C_{0}r^{\frac{4}{p+1}}\leq u^{*}(x)\leq 1-r^{\frac{4}{p+1}}$ on the unit ball, where $ C_{0}:=(λ^{*}/\barλ)^{\frac{1}{p+1}}$ and $\barλ:=\frac{8(p-1)}{(p+1)^{2}}[n-\frac{2(p-1)}{p+1}][n-\frac{4p}{p+1}]$. Our results actually complete part of the open problem which \cite{D} lef

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arXiv2011

Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

Let $λ^{*}>0$ denote the largest possible value of $λ$ such that $$ \{{array}{lllllll} Δ^{2}u=λ(1+u)^{p} & {in}\ \ \B, %0 \frac{n+4}{n-4}$ and $n$ is the exterior unit normal vector. We show that for $λ=λ^{*}$ this problem possesses a unique weak solution $u^{*}$, called the extremal solution. We prove that $u^{*}$ is singular when $n\geq 13$ for $p$ large enough, in which case $u^{*}(x)\leq r^{-\frac{4}{p-1}}-1$ on the unit ball.

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arXiv2011

Some remarks on biharmonic elliptic problems with a singular nonlinearity

We study the following semilinear biharmonic equation $$ \left\{\begin{array}{lllllll} Δ^{2}u=\fracλ{1-u}, &\quad \mbox{in}\quad \B, u=\frac{\partial u}{\partial n}=0, &\quad \mbox{on}\quad \partial\B, \end{array} \right. %\eqno(M_λ) $$ where $\B$ is the unit ball in $\R^{n}$ and $n$ is the exterior unit normal vector. We prove the existence of $λ^{*}>0$ such that for $λ\in (0,λ^{*})$ there exists a minimal (classical) solution $\underline{u}_λ$, which satisfies $0<\underline{u}_λ<1$. In the extremal case $λ=λ^{*}$, we prove the existence of a weak solution which is unique solution even in a very weak sense. Besides, several new difficulties arise and many problems still remain to be solved. we list those of particular interest in the final section.

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arXiv2010

Asymptotic behavior of positive solutions of semilinear elliptic equations in $R^{n}$

We will investigate the asymptotic behavior of positive solutions of the elliptic equation Δu+|x|^{l_{1}}u^{p}+|x|^{l_{2}}u^{q}=0 {in} R^{n}. We establish that for $n\geq 3$ and $q>p>1$, any positive radial solution of (0.1) has the following property: $\lim_{r\to\infty}r^{\frac{2+l_{1}}{p-1}}u$ and $\lim_{r\to0}r^{\frac{2+l_{2}}{q-1}}u$ always exist if $\frac{n+l_{1}}{n-2}<p<q, p\neq\frac{n+2+2l_{1}}{n-2}, q \neq\frac{n+2+2l_{2}}{n-2}.$ In addition, we prove that the singular solution of (0.1) is unique under a certain condition

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