Divergence-free Drift Research Articles

On divergence-free (form-bounded type) drifts

We develop regularity theory, including a posteriori Harnack inequality and higher integrability of the gradient of solutions, for divergence-form elliptic equations with singular divergence-free drift satisfying a broad form-boundedness-type condition (more generally, our drifts can have singular divergence). A key step in our proofs is a new iteration procedure (needed already to prove Caccioppoli's inequality), which is used in addition to the classical De Giorgi's iterations and Moser's method.

Stochastic homogenization with space-time ergodic divergence-free drift

Stochastic homogenization with space-time ergodic divergence-free drift

On shifting the principal eigenvalue of Dirichlet problem to infinity with non-transversal incompressible drift

We prove that it is always possible to add some divergence free drift vector field to some two dimensional spherical Dirichlet problem, such that the resulting principal eigenvalue lies above a prescribed bound. By construction those drift vector fields vanish on the boundary and their flow lines individually stay away from the boundary. The capacity of those drift vector fields to accelerate diffusivity originates from high frequency oscillation of the associated flow lines. The lower bounds for the spectrum are obtained through isoperimetric inequalities for flow invariant functions.

Regularity Properties of Passive Scalars with Rough Divergence-Free Drifts

Regularity Properties of Passive Scalars with Rough Divergence-Free Drifts

Large-scale regularity in stochastic homogenization with divergence-free drift

Large-scale regularity in stochastic homogenization with divergence-free drift

A partial uniqueness result and an asymptotically sharp nonuniqueness result for the Zhikov problem on the torus

We consider the stationary diffusion equation \(-\mathrm {div} (\nabla u + bu )=f\) in d-dimensional torus \(\mathbb {T}^d\), where \(f\in H^{-1}\) is a given forcing and \(b\in L^p\) is a divergence-free drift. Zhikov (Funkts Anal Prilozhen, 38(3):15–28, 2004) considered this equation in the case of a bounded, Lipschitz domain \(\Omega \subset \mathbb {R}^d\), and proved existence of solutions for \(b\in L^{2d/(d+2)}\), uniqueness for \(b\in L^2\), and has provided a point-singularity counterexample that shows nonuniqueness for \(b\in L^{3/2-}\) and \(d=3,4,5\). We apply a duality method and a DiPerna–Lions-type estimate to show uniqueness of the solutions constructed by Zhikov for \(b\in W^{1,1}\). We use a Nash iteration to demonstrate sharpness of this result, and also show that solutions in \(H^1\cap L^{p/(p-1)}\) are flexible for \(b\in L^p\), \(p\in [1,2(d-1)/(d+1))\); namely we show that the set of \(b\in L^p\) for which nonuniqueness in the class \(H^1\cap L^{p/(p-1)}\) occurs is dense in the divergence-free subspace of \(L^p\).

On the Non-diffusive Magneto-Geostrophic Equation

Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In Friedlander and Vicol (Nonlinearity 24(11)::3019–3042, 2011), the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized in frequency straight lines crossing the origin. For such well-prepared data, the local existence and uniqueness of solutions can be obtained in Sobolev spaces and the global existence holds under a size condition over the $$H^{5/2^{+}}({{\mathbb {T}}}^3)$$ norm of the perturbation.

Markov semi-groups generated by elliptic operators with divergence-free drift

In this paper we construct a conservative Markov semi-group with generator L=Δ+b⋅∇ on Rn, where b is a divergence-free vector field which belongs to L2∩Lp with n2<p. The research is motivated by the question of understanding the blow-up solutions of the fluid dynamic equations, which attracts a lot of attention in recent years.

On a maximal inequality and its application to SDEs with singular drift

In this paper we present a Doob type maximal inequality for stochastic processes satisfying the conditional increment control condition. If we assume, in addition, that the margins of the process have uniform exponential tail decay, we prove that the supremum of the process decays exponentially in the same manner. Then we apply this result to the construction of the almost everywhere stochastic flow to stochastic differential equations with singular time dependent divergence-free drift.

Remarks on Liouville type theorems for steady-state Navier–Stokes equations

Liouville type theorems for the stationary Navier-Stokes equations are proven under certain assumptions. These assumptions are motivated by conditions that appear in Liouvile type theorems for the heat equations with a given divergence free drift.

Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts

AbstractThis paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

Parabolic equations with divergence-free drift in space L_{t}^{l}L_{x}^{q}

In this paper, we study the fundamental solution $\varGamma(t,x;\tau,\xi)$ of the parabolic operator $L_t=\partial_t-\Delta+b(t,x)\cdot\nabla$, 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 $L^{\ell}(0,T;L^q(\mathbb{R}^n))$. The regularity of weak solutions to the parabolic equation $L_tu=0$ depends critically on the value of the parabolic exponent $\gamma=2/\ell+n/q$. Without the divergence-free condition on $b$, the regularity of weak solutions has been established when $\gamma\leq1$, and the heat kernel estimate has been obtained as well, except for the case that $\ell=\infty$, $q=n$. The regularity of weak solutions was deemed not true for the critical case $L^{\infty}(0,T;L^n(\mathbb{R}^n))$ for a general $b$, while it is true for the divergence-free case, and a written proof can be deduced from the results in [Yu. A. Semenov, \emph{Regularity theorems for parabolic equations}, J. Funct. Anal. \textbf{231} (2006), no. 2, 375--417]. One of the results obtained in the present paper establishes the Aronson-type estimate for critical and supercritical cases and for vector fields $b$ which are divergence free. We will prove the best possible lower and upper bounds for the fundamental solution one can derive under the current approach. The significance of the divergence-free condition has entered the study of parabolic equations rather recently, mainly because of the discovery of the compensated compactness. The interest for the study of such parabolic equations comes from its connections with Leray's weak solutions of the Navier-Stokes equations and the Taylor diffusion associated with a vector field where the heat operator $L_t$ appears naturally.

Quenched central limit theorem for random walks in doubly stochastic random environment

We prove the quenched version of the central limit theorem for the displacement of a random walk in doubly stochastic random environment, under the $H_{-1}$-condition, with slightly stronger, $\mathcal{L}^{2+\varepsilon}$ (rather than $\mathcal{L}^{2}$) integrability condition on the stream tensor. On the way we extend Nash’s moment bound to the nonreversible, divergence-free drift case, with unbounded ($\mathcal{L}^{2+\varepsilon}$) stream tensor. This paper is a sequel of [Ann. Probab. 45 (2017) 4307–4347] and relies on technical results quoted from there.

Solutions to a Class of Forced Drift-Diffusion Equations with Applications to the Magneto-Geostrophic Equations

We prove the global existence of classical solutions to a class of forced drift-diffusion equations with $$L^2$$ initial data and divergence free drift velocity $$\{u^\nu \}_{\nu _\ge 0}\subset L^\infty _t BMO^{-1}_x$$ , and we obtain strong convergence of solutions as the viscosity $$\nu $$ vanishes. We then apply our results to a family of active scalar equations which includes the three dimensional magneto-geostrophic $$\{\hbox {MG}^\nu \}_{\nu \ge 0}$$ equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth’s fluid core. We prove the existence of a compact global attractor $$\{\mathcal {A}^\nu \}_{\nu \ge 0}$$ in $$L^2(\mathbb {T}^3)$$ for the $$\hbox {MG}^\nu $$ equations including the critical equation where $$\nu =0$$ . Furthermore, we obtain the upper semicontinuity of the global attractor as $$\nu $$ vanishes.

On the uniqueness of solutions to stationary convection–diffusion equations with generalized divergence-free drift

Let A be a skew-symmetric matrix in , — a bounded Lipschitz domain in , . The Dirichlet problem , , has at least one solution obtained by approximating A and passing to the limit. In 2004 V. V. Zhikov constructed an example of nonuniqueness. In the same paper he proved the uniqueness of solutions if the norms of A are o(p) as p goes to infinity. We prove the uniqueness of solutions if for some , which generalizes Zhikov’s theorem.

Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation (−Δ)α in the regime α∈(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∈(0,1), the Cβ norm of the solution depends only on the size of the drift in critical spaces of the form Ltq(BMOx−γ) with q>2 and γ∈(0,2α−1], along with the Lx2 norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.

On optimal Hölder regularity of solutions to the equation Δu + b ⋅ ∇u = 0 in two dimensions

We show that for an L2 drift b in two dimensions, if the Hardy norm of divb is small, then the weak solutions to Δu+b⋅∇u=0 have the same optimal Hölder regularity as in the case of divergence-free drift, that is, u∈Clocα for all α∈(0,1).

Existence and uniqueness of weak solutions for a class of fractional superdiffusion equations

In this paper, we consider the existence and uniqueness of weak solutions for a class of fractional superdiffusion equations with initial-boundary conditions. For a multidimensional fractional drift superdiffusion equation, we just consider the simplest case with divergence-free drift velocity u in L^{2}(Omega) only depending on the spatial variable x. Finally, exploiting the Schauder fixed point theorem combined with the Arzelà-Ascoli compactness theorem, we obtain the existence and uniqueness of weak solutions in the standard Banach space C([0,T]; H_{0}^{1}(Omega)) for a class of fractional superdiffusion equations.

The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity

ABSTRACTWe establish the Harnack inequality for advection-diffusion equations with divergence-free drifts by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale invariant spaces.

On the Smoothing Effect in the Kinematic Dynamo Equations in Critical Spaces

We study the smoothing effects of diffusion in a 3D vector system of linear advection-diffusion PDEs known as the kinematic dynamo equations. We examine the case where both the divergence free drift velocity \({\mathbf{u}(x,t)}\) and the mild solution \({\mathbf{B}(x,t)}\) to the PDE obey the natural scaling of the equation. We consider initial data \({\mathbf{B}_0(x) \in L^3(\mathbb{R}^3)}\) and demonstrate how the regularization effects of diffusion are influenced by the profile of \({\mathbf{u}(x,t)}\).