We show the existence and uniqueness as well as boundedness of weak solutions to linear elliptic equations with $L^2$-drifts of negative divergence and singular zero-order terms which are positive. Our main target is to show the $L^r$-contraction properties of the unique weak solutions. Indeed, using the Dirichlet form theory, we construct a sub-Markovian $C_0$-resolvent of contractions and identify it to the weak solutions. Furthermore, we derive an $L^1$-stability result through an extended version of the $L^1$-contraction property.
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.
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…
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…
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…
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.
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…
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 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 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…
Kinetic Fokker--Planck Equations with Drifts in a Supercritical Range
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Department of Mathematics Sciences, Graduate School of Science, Kyoto University（京都大学理学研究科数学科学系）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
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 prove the exponential convergence to the equilibrium, quantified by Rényi divergence, of the solution of the Fokker-Planck equation with drift given by the gradient of a strictly convex potential. This extends the classical exponential decay result on the relative entropy for the same equation.
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.
Existence of weak solutions for nonlinear drift-diffusion equations with measure data
Chungbuk National University（忠北国立大学）；Yonsei University（延世大学）；Hannam University（韩南大学）；Korea University of Technology and Education（韩国科学技术院）；Korea Institute for Advanced Study（韩国高等研究院）
This paper considers a class of nonlinear, degenerate drift- diffusion equations. We study well-posedness and regularity properties of the solutions, with the goal to achieve uniform Hölder regularity in terms of $L^p$-bound on the drift vector field. A formal scaling argument yields that the threshold for such estimates is $p=d$, while our estimates are for $p>d+\frac{4}{d+2}$. On the other hand we are able to show by a series of examples that one needs $p>d$ for such estimates, even for divergence free drift.
We illustrate some novel contraction and regularizing properties of the Heat flow in metric-measure spaces that emphasize an interplay between Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorvich distances. Contraction properties of Hellinger-Kakutani distances and general Csiszár divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of the flow. When weaker transport distances are involved, we will show that contraction and regularizing effects rely on the dual formulations of the distances and are strictly related to lower Ricci c…
We consider Dirichlet problems for linear elliptic equations of second order in divergence form on a bounded or exterior smooth domain $Ω$ in $\mathbb{R}^n$, $n \ge 3$, with drifts $\mathbf{b}$ in the critical weak $L^n$-space $L^{n,\infty}(Ω; \mathbb{R}^n )$. First, assuming that the drift $\mathbf{b}$ has nonnegative weak divergence in $L^{n/2, \infty }(Ω)$, we establish existence and uniqueness of weak solutions in $W^{1,p}(Ω)$ or $D^{1,p}(Ω)$ for any $p$ with $n' = n/(n-1)< p < n$. By duality, a similar result also holds for the dual problem. Next, we prove $W^{1,n+\varepsilon}$ or $W^{2, …
We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-Δu + b^{(α)}\cdot \nabla u= f$ in a bounded domain $Ω\subset {\mathbb R^2}$ containing the origin, where $f \in W^{-1}_q(Ω) $ with $q>2$ and $b^{(α)}:=b-α\frac{x}{|x|^2}$, $b$ is a divergence-free vector field and $α\in {\mathbb R}$ is a parameter.
