Damir Kinzebulatov, Reihaneh Vafadar
We develop regularity theory for elliptic Kolmogorov operator with divergence-free drift in a large class (or, more generally, drift having singular divergence). A key step in our proofs is "Caccioppoli's iterations", used in addition to the classical De Giorgi's iterations and Moser's method.
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We prove well-posedness for a class of second-order SPDEs with multiplicative Wiener noise and doubly nonlinear drift in divergence form involving maximal monotone graphs defined everywhere, on which neither growth nor continuity assumptions are imposed.
We provide a simple proof of quenched stochastic homogenization for random environments with a mean zero, divergence-free drift under the assumption that the drift admits a stationary $L^d$-integrable stream matrix in $d\geq 3$ or an $L^{2+δ}$-integrable stream matrix in $d=2$. In addition, we prove that the environment almost surely satisfies a large-scale Hölder regularity estimate and first-order Liouville principle.
We prove that diffusion equations with a space-time stationary and ergodic, divergence-free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uni…
We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, an…
In this paper we establish the uniqueness of a solution to a stationary convection-diffusion equation in divergence form with an exponentially summable generalized divergence-free drift.
We prove a local energy identity for a class of distributional solutions, in $L_{2,\infty} \cap W^{1,0}_2$, of parabolic equations with divergence-free drift.
We consider a degenerate wave equation with drift in presence of a leading operator which is not in divergence form. We provide some conditions for the boundary controllability of the associated Cauchy problem.
We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form $\operatorname{div} γ(\nabla \cdot)$, where $γ$ is a maximal monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that $γ$ is single-valued, a solution with better integrability and regularity prop…
On the periodic homogenization of elliptic equations in non-divergence form with large drifts
本文研究带大漂移项的非散度型周期椭圆方程的定量均匀化，通过中心化条件与不变测度将其转化为无漂移的散度型方程，从而获得收敛速率与一致Lipschitz正则性。
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…
Linear stabilization for a degenerate wave equation in non divergence form with drift
本文研究带漂移项的非散度型一维退化波动方程，通过在退化端点施加齐次Dirichlet边界条件并在另一端施加边界阻尼，给出了Cauchy问题解一致指数衰减的充分条件。
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.
This paper investigates quantitative estimates in elliptic homogenization of non-divergence form with unbounded drift and an interface, which continues the study of the previous work by Hairer and Manson [Ann. Probab. 39(2011) 648-682], where they investigated the limiting long time/large scale behavior of such a process under diffusive rescaling. We determine the effective equation and obtain the size estimates of the gradient of Green functions as well as the optimal (in general) convergence rates. The proof relies on transferring the non-divergence form into the divergence-form with the coe…
We are interested in some properties related to the solutions of non-local diffusion equations with divergence free drift. Existence, maximum principle and a positivity principle are proved. In order to study Holder regularity, we apply a method that relies in the Holder-Hardy spaces duality and in the molecular characterisation of local Hardy spaces. In these equations, the diffusion is given by Lévy-type operators with an associated Lévy measure satisfying some upper and lower bounds.
We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t - Δ+b\cdot\nabla$ and $-Δ+b\cdot\nabla$ with a divergence-free drift $b$. We prove the Liouville theorem and Harnack inequality when $b\in L_\infty(BMO^{-1})$ resp. $b\in BMO^{-1}$ and provide a counterexample to such results demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BM…
Existence of weak solutions for fast diffusion equation with a divergence type of drift term
在漂移项满足可积性条件下，构造满足L^q能量不等式和Wasserstein空间速度估计的非负弱解；当漂移项无散时，可放宽条件，并应用于粘性Boussinesq系统。
We consider the problem of constructing weak solutions to the Itô and to the Stratonovich stochastic differential equations having critical-order singularities in the drift and critical-order discontinuities in the dispersion matrix.
For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are Hölder continuous. We construct time-independent supercritical drifts in $L^{n-λ}(\mathbb{R}^n)$ with arbitrarily small $λ>0$ such that the Harnack inequality and the Hölder continuity fail in both the elliptic and the parabolic cases, thus confirming a conjecture by Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimen…
Form-boundedness and SDEs with singular drift
该研究梳理并改进了带奇异漂移的随机微分方程在若干极小假设下的弱适定性与强适定性相关研究结果。
Interior pointwise $C^α$ regularity for elliptic and parabolic equations with divergence-free drifts
本文针对带无散漂移项的椭圆与抛物方程，利用能量不等式和扰动技术，在三类小漂移条件下证明弱解对任意 $0<α<1$ 的内部逐点 $C^α$ 正则性。
We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation \[ \partial_t θ- Δθ+ b \cdot \nabla θ= 0 \] to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L^q_t L^p_x$, where $\frac{2}{q} + \frac{n}{p} < 2$, or $L^p_x L^q_t$, where $\frac{3}{q} + \frac{n-1}{p} < 2$. For steady drifts, the condition reduces to $b \in L^{\frac{n-1}{2}+}$. The space $L^1_t L^\infty_x$ of drifts with `bounded total speed' is a borderl…
Strong solutions of SDEs with singular (form-bounded) drift via Roeckner-Zhao approach
本文采用Roeckner-Zhao方法，在最小假设下证明了具有奇异漂移的随机微分方程的强适定性。
Given two elliptic operators L and M in nondivergence form, with coefficients A_L(x), A_M(x) and drift terms b_L(x), b_M(x), respectively, satisfying a Carleson measure disagreement condition in a Lipschitz domain Omega in R^{n+1}, then their harmonic measures are mutually absolutely continuous. As an application of this, a new approximation argument and known results we obtain necessary and sufficient conditions for a single operator L (in divergence or nondivergence form) to have regular harmonic measure with respect to Lebesgue measure. The results are sharp in all cases.
We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift \[-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)\] in some domain $Q\subset \mathbb{R}^{n+1}$. We prove growth theorems and the interior Harnack inequality as the main results. In this paper, we will only assume the drift $b$ is in certain Lebesgue spaces which are critical under the parabolic scaling but not necessarily to be bounded. In the last section, some applications of the interior Harnack inequality are presented. In particular, we show there is…
Heat equation with singular drift which has bounded solution discontinuous at an interior point
本文构造了一个带奇异漂移的一维热方程，其光滑初值柯西问题的解有界但在内部公共点不连续。
We discuss the local properties of weak solutions to the equation $-Δu + b\cdot\nabla u=0$. The corresponding theory is well-known in the case $b\in L_n$, where $n$ is the dimension of the space. Our main interest is focused on the case $b\in L_2$. In this case the structure assumption $\operatorname{div} b=0$ turns out to be crucial.
We consider uniformly parabolic equations and inequalities of second order in the non-divergence form with drift \[-u_{t}+Lu=-u_{t}+\sum_{ij}a_{ij}D_{ij}u+\sum b_{i}D_{i}u=0\,(\geq0,\,\leq0)\] in some domain $Ω\subset \mathbb{R}^{n+1}$. We prove a variant of Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate with $L^{p}$ norm of the inhomogeneous term for some number $p<n+1$. Based on it, we derive the growth theorems and the interior Harnack inequality. In this paper, we will only assume the drift $b$ is in certain Morrey spaces defined below which are critical under the parabolic scaling but not…
Existence of weak solutions for Porous medium equation with a divergence type of drift term in a bounded domain
该研究将全空间下的相关结论推广到有界区域场景，证明了带散度型漂移项、无通量边界的多孔介质方程在Wasserstein空间中L^q弱解的存在性，并将结果应用于消耗型Keller-Segel方程的弱解构造。
This work focuses on drift-diffusion equations with fractional dissipation $(-Δ)^α$ in the regime $α\in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $β\in (0,1)$, the $C^β$ norm of the solution depends only on the size of the drift in critical spaces of the form $L^{q}_{t}(BMO^{-γ}_{x})$ with $q>2$ and $γ\in (0,2α-1]$, along with the $L^{2}_{x}$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.
We consider degenerate porous medium equations with a divergence type of drift terms. We establish the existence of $L^{q}$-weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on $q$ and $m$ caused by the nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider if the drift term is divergence-free. Similar conditions of gradients of drift…
Well-posedness of linear elliptic equations with $L^d$-drifts under divergence-type conditions
本文研究了具有临界阶漂移项的线性椭圆方程的适定性，通过散度自由变换和漂移向量场的结构条件，放宽了零阶系数的正则性要求，证明了弱解的存在性和唯一性。
Interior Harnack inequality and Hölder estimates for linearized Monge-Ampère equations in divergence form with drift
本文研究了带漂移项的散度形式线性化 Monge-Ampère 方程解的内部估计，证明了二维及高维（需附加 Hessian 可积性假设）情形下的内部 Harnack 不等式与 Hölder 估计，推广了 Le 的相关结果。
On the contraction properties for weak solutions to linear elliptic equations with $L^2$-drifts of negative divergence
本文研究具有负散度 $L^2$-漂移项和正奇异零阶项的线性椭圆方程，利用 Dirichlet 型理论构造次 Markov $C_0$-收缩预解式，证明弱解的存在唯一性、有界性及 $L^r$-收缩性质，并导出 $L^1$-稳定性结果。
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…
Weak well-posedness of energy solutions to singular SDEs with supercritical distributional drift
该研究针对d≥2维空间中带加性噪声与分布漂移的奇异SDE，运用能量解与奇异SPDE生成元新思路，在超临界情形下证明了时变无散漂移及满足结构假设的时不变漂移下能量解的弱适定性，拓展了适定性成立的漂移奇异性范围。
Asymptotic Behavior of the Principal Eigenvalue Problems with Large Divergence-Free Drifts
本文研究了大散度无漂移漂移问题的主特征值问题的渐进行为，证明了主特征对的收敛性并分析了其极限轮廓。
We show the $L^r(\mathbb{R}^d, μ)$-uniqueness for any $r \in (1, 2]$ and the essential self-adjointness of a Dirichlet operator $Lf = Δf +\langle \frac{1}ρ\nabla ρ, \nabla f \rangle$, $f \in C_0^{\infty}(\mathbb{R}^d)$ with $d \geq 3$ and $μ=ρdx$. In particular, $\nabla ρ$ is allowed to be in $L^d_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ or in $L^{2+\varepsilon}_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ for some $\varepsilon>0$, while $ρ$ is required to be locally bounded below and above by strictly positive constants. The main tools in this paper are elliptic regularity results for divergence and non-dive…
Divergence-free drifts decrease concentration
本文研究了无散漂移场对对流-扩散方程解集中度的影响，发现其能降低绝对连续性模量，且在特定条件下导致解的方差、熵和$L^p$范数变化。
Analysis of linear elliptic equations with general drifts and $L^1$-zero-order terms
本文研究了具有一般漂移项和 $L^1$-零阶项的线性椭圆方程 Dirichlet 边值问题，通过将漂移项转化为弱无散度形式，证明了弱解和强解的存在唯一性，并指出估计常数可显式计算以应用于误差分析。
Pointwise well-posedness results for degenerate Itô-SDEs with locally bounded drifts
本文基于前人研究，探讨了退化伊藤随机微分方程在局部有界漂移下的点wisewell-posedness问题，通过椭圆正则性结果和额外正则性假设，提供欧几里得空间中每个起始点的点wise分析，主要贡献是弱well-posedness，即弱存在性和唯一性。
Asymptotic estimates for solutions of inhomogeneous non-divergence diffusion equations with drifts
本文研究了非齐次非散度扩散方程解的渐近行为，通过分析漂移项和数据的渐近特性，得出了解的渐近估计和收敛性结果。
We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+λu=f+\mathrm{div } \mathbf{F}\quad \text{in } Ω\quad\text{and}\quad u=0\quad \text{on } \partialΩ, \end{equation*} in bounded Lipschitz domain $Ω$ in $\mathbb{R}^2$, where $A:\mathbb{R}^2\rightarrow \mathbb{R}^{2^2}$, $\mathbf{b} : Ω\rightarrow \mathbb{R}^2$, and $λ\geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $Ω$ has small Lipschitz constant, and $\mathrm{div } A,\,\mathbf{b} \in L^{2}(Ω;…
We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term $B$ and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato, Flandoli, Priola and M. Rockner, Annals of Prob., published online in 2012) which generalized Veretennikov's fundamental result to infinite dimensions assuming boundedness of the drift term. As in our previous paper pathwise uniqueness holds for a large class, but not for every initial condition. We also include an applica…
We construct fundamental solutions of second-order parabolic systems of divergence form with bounded and measurable leading coefficients and divergence free first-order coefficients in the class of $BMO^{-1}_x$, under the assumption that weak solutions of the system satisfy a certain local boundedness estimate. We also establish Gaussian upper bound for such fundamental solutions under the same conditions.
Existence and regularity for perturbed Stokes system with critical drift in 2D
本文研究了二维有界利普希茨域中临界无散漂移扰动下的斯托克斯系统，证明了在弱L²空间中大漂移下的q-弱解存在性及在L²空间中任意大L下的W^{1,2}解存在性。
Elliptic theory and dimension drop for operators with divergence-free form-bounded drifts in one-sided chord-arc domains
本文针对单侧弦弧域中带形式有界漂移的椭圆算子，建立边值与位势理论，推导相关估计与性质，最终利用AI构造出具有维数下降调和测度的特定算子。
We add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the behavior of the principal eigenvalue for the Dirichlet problem. The eigenvalue remains bounded if and only if the drift has non-trivial first integrals in the domain of the quadratic form of the fractional Laplacian.
Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients
本文研究了双散度型二阶抛物方程非负解的Harnack不等式，设定主系数满足Dini平均振荡条件且低阶系数属于Morrey类，推进了相关方程及Fokker-Planck-Kolmogorov方程的理论基础。
Kinetic Fokker--Planck Equations with Drifts in a Supercritical Range
该研究针对各向异性混合勒贝格空间中带粗糙无散漂移项的动力学福克-普朗克方程，结合能量抵消、各向异性正则化估计与De Giorgi迭代，建立弱次解局部有界性并证明超临界区域柯西问题整体适定性。
Quantitative propagation of chaos for particle systems with bounded kernels and multiplicative noise
研究在漂移和扩散系数中均存在相互作用的随机粒子系统，证明在漂移核有界的情况下，定量混沌传播的成立，扩展了Jabin和Wang的指数大数定律到扩散核，并处理熵演化方程中乘性噪声带来的误差项。
Existence and Blow-Up for Nonlinear Fokker-Planck with Controlled Drift Divergence
该研究将相关文献的临界参数整体存在性结果推广到特定非自治场条件下的多维非线性Fokker-Planck方程，采用抛物对称化方法完成证明。
Let $Ω$ be an arbitrary bounded domain of $\R^n$. We study the right invertibility of the divergence on $Ω$ in weighted Lebesgue and Sobolev spaces on $Ω$, and rely this invertibility to a geometric characterization of $Ω$ and to weighted Poincaré inequalities on $Ω$. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when $Ω$ is Lipschitz or, more generally, when $Ω$ is a John domain, and focus on the case of $s$-John domains.
We provide a vast class of counterexamples to the chain rule for the divergence of bounded vector fields in three space dimensions. Our convex integration approach allows us to produce renormalization defects of various kinds, which in a sense quantify the breakdown of the chain rule. For instance, we can construct defects which are absolutely continuous with respect to Lebesgue measure, or defects which are not even measures.
In this paper we study the fundamental solution $\varGamma(t,x;τ,ξ)$ of the parabolic operator $L_{t}=\partial_{t}-Δ+b(t,x)\cdot\nabla$, where for every $t$, $b(t,\cdot)$ is a divergence-free vector field, and we consider the case that $b$ belongs to the Lebesgue space $L^{l}\left(0,T;L^{q}\left(\mathbb{R}^{n}\right)\right)$. The regularity of weak solutions to the parabolic equation $L_{t}u=0$ depends critically on the value of the parabolic exponent $γ=\frac{2}{l}+\frac{n}{q}$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $γ\leq1$, …
We introduce a family of pairings between a bounded divergence-measure vector field and a function $u$ of bounded variation, depending on the choice of the pointwise representative of $u$. We prove that these pairings inherit from the standard one, introduced in [6,10], all the main properties and features (e.g. coarea, Leibniz and Gauss--Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in $BV$. We remark that the standard pairing in general does not share this property.
We construct a strong Feller process associated with $-Δ+ σ\cdot \nabla$, with drift $σ$ in a wide class of measures (weakly form-bounded measures, e.g. combining weak $L^d$ and Kato class measure singularities), by exploiting a quantitative dependence of the smoothness of the domain of an operator realization of $-Δ+ σ\cdot \nabla$ generating a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$, $p>d-1$, on the value of the form-bound of $σ$. Our method admits extension to other types of perturbations of $-Δ$ or $(-Δ)^{\fracα{2}}$, e.g. to yield new $L^p$-regularity results for Schrödinger ope…
Logarithmic Sobolev Inequalities for Bounded Domains and Applications to Drift-Diffusion Equations
本文提出新型 Gagliardo-Nirenberg 型插值不等式，据此建立高维有界光滑区域上的对数 Sobolev 不等式，并用于证明非线性非局部漂移-扩散模型解指数收敛至常数稳态。
Potential theory for nonlocal drift-diffusion equations
本文通过逐点位势估计证明了非局部漂移-扩散方程的精细正则性结果，仅需对无散度漂移项作极小假设，并推导出耗散表面准地转方程的新估计。
$BV$-Estimates for Non-Linear Parabolic PDE with Linear Drift
本文研究了含线性漂移项的非线性抛物型PDE解的BV正则性，探讨了有界域中混合边界条件下的解的稳定性及线性输运方程的特殊情况。
Boundary controllability for degenerate/singular hyperbolic equations in nondivergence form with drift
该研究针对带漂移项的非散度型退化/奇异波动方程的零能控问题，借助能量方法与边界能观性，给出了局域于非退化边界点的控制下边界能控性的相关条件。
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