Transport equation: extension of classical results for div $b\in$ BMO

Elliptic equations in divergence form with partially BMO coefficients

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients $a^{ij}$ are assumed to be measurable in one direction and have small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that $a^{ij}$ have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We …

$L^{q}$ estimates for nonlocal p-Laplacian type equations with BMO kernel coefficients in divergence form

We study $s$-fractional $p$-Laplacian type equations with discontinuous kernel coefficients in divergence form to establish $W^{s+σ,q}$ estimates for any choice of pairs $( σ,q)$ with $q\in(p,\infty)$ and $σ\in\left(0,\min\left\{\frac{s}{p-1},1-s\right\}\right)$ under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a non-homogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in th…

BMO solutions to quasilinear equations of $p$-Laplace type

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $-Δ_{p} u = μ$ in $\mathbb{R}^n$, $u\ge 0$, where $μ$ is a locally finite Radon measure, and $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian ($p>1$). We also characterize BMO solutions to equations $-Δ_{p} u = σu^{q} + μ$ in $\mathbb{R}^n$, $u\ge 0$, with $q>0$, where both $μ$ and $σ$ are locally finite Radon measures. Our main results hold for a class of more general quasilinear operators ${\rm div}(\mathcal{A}(x, \nabla \cdot))$ in place of $Δ_{p}$.

Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

In this paper, we prove the Lorentz space $L^{q,p}$-estimates for gradients of very weak solutions to the linear parabolic equations with $\mathbf{A}_q$-weights $$u_t-\operatorname{div}(A(x,t)\nabla u)=\operatorname{div}(F),$$ in a bounded domain $Ω\times (0,T)\subset\mathbb{R}^{N+1}$, where $A$ has a small mean oscillation, and $Ω$ is a Lipchistz domain with a small Lipschitz constant.

Solvability of second-order equations with hierarchically partially BMO coefficients

By using some recent results for divergence form equations, we study the $L_p$-solvability of second-order elliptic and parabolic equations in nondivergence form for any $p\in (1,\infty)$. The leading coefficients are assumed to be in locally BMO spaces with suitably small BMO seminorms. We not only extend several previous results by Krylov and Kim [14]-[18] to the full range of $p$, but also deal with equations with more general coefficients.

A Note on Transport Equation in Quasiconformally Invariant Spaces

In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel-Lizorkin spaces.

Lipschitz 域上二阶线性椭圆方程的全局 BMO-Sobolev 估计

Global BMO-Sobolev Estimates for Second-Order Linear Elliptic Equations on Lipschitz Domains

本文在极小正则性假设下，利用 BMO 空间的逐点乘子刻画，建立了 Lipschitz 域上二阶线性椭圆方程弱解的全局 BMO 梯度估计，并由此得到 Hardy 空间数据对应的 $L^1$ 全局估计。

临界椭圆方程解的$BMO$与梯度估计

$BMO$ and gradient estimates for solutions of critical elliptic equations

该研究将有界$β$维平均振荡空间应用于临界椭圆方程正则性理论，改进了泊松方程与$n$-拉普拉斯方程的$BMO$正则性结果，并得到无散右端项时$n$-拉普拉斯方程的梯度$L^n$估计。

$L^p$ theory for the square roots and square functions of elliptic operators having a BMO anti-symmetric part

We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$ matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in $L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to the space $BMO(\mathbb{R}^n)$, $n\ge2$. We prove the Gaussian estimates for the kernel of $e^{-tL}$, as well as that of $\partial_t^le^{-tL}$, for any $l\in\mathbb{N}$. We show that the square root of $L$ satisfies the $L^p$ estimates $\left\Vert{L^{1/2}f}\right\Vert_{L^p}\lesssim\left\Vert{\nabla f}\right\Vert_{L^p}$ for $1<p<\infty$, and $\left\Vert{\nabla f}\right\Vert_{L^p}…

具有BMO反对称部分的二阶椭圆算子的微扰理论

Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part

本文研究带无界BMO反对称部分的散度型椭圆算子的$L^p$狄利克雷问题微扰理论，证明系数差满足Carleson条件时可解性的保持与阶数关系，并将其应用于有界Lipschitz域的相关问题。

The energy conservation and regularity for the Navier-Stokes equations

In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $\lim_{t\to T}\sqrt{T-t}||u(t)||_{BMO}<\infty$ and $\lim_{t\to T}\sqrt{T-t}||u(t)||_{L^\infty}=\infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $u\in L^{2,\infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,\infty}([0,T];BMO(Ω))}<\infty$ for arbitrary $Ω\subseteq\mathbb{R}^3$ then the local energy equ…

Parabolic equations with singular divergence-free drift vector fields

In this paper, we study an elliptic operator in divergence-form but not necessary symmetric. In particular, our results can be applied to elliptic operator $L=νΔ+u(x,t)\cdot\nabla$, where $u(\cdot,t)$ is a time-dependent vector field in $\mathbb{R}^{n}$, which is divergence-free in distribution sense, i.e. $\nabla\cdot u=0$. Suppose $u\in L_{t}^{\infty}(\textrm{BMO}_{x}^{-1})$. We show the existence of the fundamental solution $\varGamma(x,t;ξ,τ)$ of the parabolic operator $L-\partial_{t}$, and show that $\varGamma$ satisfies the Aronson estimate with a constant depending only on the dimension…

通过外推法证明二维欧拉方程与输运方程解的唯一性

Uniqueness for 2D Euler and transport equations via extrapolation

本文利用外推理论建立新框架证明输运方程解的唯一性，并统一推广Yudovich与Vishik的二维欧拉方程经典结果，将唯一性推广到包含Yudovich空间和BMO的新函数空间尺度。

A note on local $W^{1,p}$-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts

We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted $L^p$-estimates for the gradients of weak solutions. Our results improve the classical one to the borderline case by replacing the $L^\infty$-assumption on solutions by solutions in the John-Nirenberg \textup{BMO} space. The results are also generalized to parabolic equations in divergence form with small oscillation elliptic symmetric coefficients and therefore imp…

Yudovich类中2D Euler方程的最优正则性

Optimal Regularity for the 2D Euler Equations in the Yudovich class

本文研究了由具有BMO梯度的速度场驱动的输运方程的最优正则性，并将其应用于初始涡度有界情形下的2D Euler方程，通过Bahouri和Chemin涡旋面片例子的变体证明了结果的尖锐性。

Ill-posedness for the Navier-Stokes equations in critical Besov spaces $\dot B^{-1}_{\infty,q}$

We study the Cauchy problem for the incompressible Navier-Stokes equation \begin{align} u_t -Δu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= δu_0. \label{NS} \end{align} For arbitrarily small $δ>0$, we show that the solution map $δu_0 \to u$ in critical Besov spaces $\dot B^{-1}_{\infty,q}$ ($\forall \ q\in [1,2]$) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in $BMO^{-1}$. Taking notice of the embedding $\dot B^{-1}_{\infty,q} \subset BMO^{-1}$ ($q\le 2$), we see that for sufficiently small $δ>0$, $u_0\in \dot B^{…

具有无界散度的可压缩非牛顿流体 Stokes 系统的弱解

Weak solutions for the Stokes system for compressible non-Newtonian fluids with unbounded divergence

本文研究了小 Reynolds 数下可压缩非牛顿流体偏微分方程系统的弱解存在性，在速度场散度缺乏 $L^\infty$ 估计的情况下，结合奇异算子正则性理论与 $BMO$ 函数对数积分不等式构造了弱解。

频率局部化 Koch-Tataru 空间中小初值 Navier-Stokes 方程的整体适定性

Global well-posedness of the Navier-Stokes equations for small initial data in frequency localized Koch-Tataru's space

研究三维不可压Navier-Stokes方程小初值整体适定性，提出高频在BMO^{-1}、低频在Besov空间分别小的频率局部化条件，构造强L2取初值的整体光滑解，弱化Koch-Tataru假设并保证能量平衡。

Global Calderón--Zygmund theory for parabolic $p$-Laplacian system: the case $1<p\leq \frac{2n}{n+2}$

The aim of this paper is to establish global Calderón--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \operatorname{div} (|F|^{p-2}F)~\text{in}~Ω\times (0,T)\subset \mathbb{R}^{n+1}, $$ proving that $$F\in L^q\Rightarrow \nabla u\in L^q,$$ for any $q>\max\{p,\frac{n(2-p)}{2}\}$ and $p>1$. Acerbi and Mingione \cite{Acerbi07} proved this estimate in the case $p>\frac{2n}{n+2}$. In this article we settle the case $1<p\leq \frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm…

Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition

We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potent…

Global Sobolev regularity for general elliptic equations of $p$-Laplacian type

We derive global gradient estimates for $W^{1,p}_0(Ω)$-weak solutions to quasilinear elliptic equations of the form $$ \mathrm{div\,}\mathbf{a}(x,u,Du)=\mathrm{div\,}(|F|^{p-2}F) $$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and Hölder continuous in $u.$ In the case when $p\geq n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonl…

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Consider the nonlinear parabolic equation in the form $$ u_t-{\rm div} \mathbf{a}(D u,x,t)={\rm div}\,(|F|^{p-2}F) \quad \text{in} \quad Ω\times(0,T), $$ where $T>0$ and $Ω$ is a Reifenberg domain. We suppose that the nonlinearity $\mathbf{a}(ξ,x,t)$ has a small BMO norm with respect to $x$ and is merely measurable and bounded with respect to the time variable $t$. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates ex…

On higher integrability estimates for elliptic equations with singular coefficients

In this note we establish existence and uniqueness of weak solutions of linear elliptic equation $\text{div}[\mathbf{A}(x) \nabla u] = \text{div}{\mathbf{F}(x)}$, where the matrix $\mathbf{A}$ is just measurable and its skew-symmetric part can be unbounded. Global reverse Hölder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of $\mathbf{A}$ is not sufficient for higher integrability estimates even when the symmetric part of $\mathbf{A}$ is the identity matrix. In addition, the exampl…

Tent space well-posedness for parabolic Cauchy problems with rough coefficients

We study the well-posedness of Cauchy problems on the upper half space $\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u =(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly elliptic coefficients. We address initial data lying in $L^p$ ($1<p<\infty$) and $BMO$ ($p=\infty$) spaces and work with weak solutions. Our main result is the identification of a new well-posedeness class, given for $p\in(1,\infty]$ by distributions satisfying $\nabla^m u \in T^{p,2}_m$, where $T^{p,2}_m$ is a parabolic version of the tent space of Coifman--Meyer--Stein. In the r…

Stochastic transport equations with unbounded divergence

We study in this article the existence and uniqueness of solutions to a class of stochastic transport equations with irregular coefficients and unbounded divergence. In the first result we assume the drift is $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ and the divergence is the locally integrable. In the second result we show that the smoothing acts as a selection criterion when the drift is in $L^{2}([0,T] \times \R^{d})\cap L^{\infty}([0,T] \times \R^{d})$ without any condition on the divergence.

Regularity for a quasilinear continuous casting problem

In this paper study the regularity of continuous casting problem \begin{equation} \hbox{div}(|\nabla u|^{p-2}\nabla u-{\bf v} β(u))=0\tag{$\sharp$} \end{equation} for prescribed constant velocity $\bf v$ and enthalpy $β(u)$ with jump discontinuity at $u=0$. We establish the following estimates: local log-Lipschitz $p>2$ for $u$ (and BMO for $\nabla u$) for two phase, Lipschitz $p>1$ for one phase and linear growth up-to boundary near the contact points. We also prove that the free boundary is continuous curve in the direction of $\bf v$ in two spatial dimensions. The proof is based on a delica…

SPDEs驱动的分歧自由传输噪声下的遍历性

Ergodicity for SPDEs driven by divergence-free transport noise

本文研究了由分歧自由传输噪声驱动的SPDEs的遍历性，证明强混合噪声可确保唯一不变测度，即使确定性部分存在多个稳态。

Linear transport equations for vector fields with subexponentially integrable divergence

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline integrability assumptions on the divergence of the velocity field $b$. For $W^{1,1}_{loc}$ vector fields $b$ satisfying $\frac{|b(x,t)|}{1+|x|}\in L^1(0,T; L^1)+L^1(0,T; L^\infty)$ and $$\operatorname{div} b\in L^1(0,T;L^\infty) + L^1\left(0,T; \operatorname{Exp}\left(\frac{L}{\log L}\right)\right),$$ we prove existence and uniqueness of weak solutions. Moreover, …

Existence and uniqueness of weak solutions of the compressible spherically symmetric Navier-Stokes equations

One of the most influential fundamental tools in harmonic analysis is Riesz transform. It maps $L^p$ functions to $L^p$ functions for any $p\in (1,\infty)$ which plays an important role in singular operators. As an application in fluid dynamics, the norm equivalence between $\|\nabla u\|_{L^p}$ and $\|\mbox{div} u\|_{L^p}+\|\mbox{curl} u\|_{L^p}$ is well established for $p\in (1,\infty)$. However, since Riesz operators sent bounded functions only to BMO functions, there is no hope to bound $\|\nabla u\|_{L^\infty}$ in terms of $\|\mbox{div} u\|_{L^\infty}+\|\mbox{curl} u\|_{L^\infty}$. As poin…

Smoothing does not give a selection principle for transport equations with bounded autonomous fields

We give an example of a bounded divergence free autonomous vector field in $\mathbb R^3$ (and of a nonautonomous bounded divergence free vector field in $\mathbb R^2$) and of a bounded initial data for which the Cauchy problem for the corresponding transport equation has $2$ distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory

We discuss recent advances in the theory of quasilinear equations of the type $ -Δ_{p} u = σu^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $σ$ is a nonnegative measurable function, or measure, for the $p$-Laplacian $Δ_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$, as well as more general quasilinear, fractional Laplacian, and Hessian operators. Within this context, we obtain some new results, in particular, necessary and sufficient conditions for the existence of solutions $u \in \text{BMO}(\mathbb{R}^n)$, $u \in L^r_{\rm loc}(\mathbb{R}^n)$, etc., and prove an enhanced…

Flows for non-smooth vector fields with subexponentially integrable divergence

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna-Lions [DPL89]. A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions.