Transport and continuity equations with (very) rough noise

Generalized Burgers equation with rough transport noise

We introduce a new technique for studying well posedness and energy estimates for evolution equations with a rough transport term. The technique is based on finding suitable space-time weight functions for the equations at hand. As an example we study the well posedness of the generalized viscous Burgers equation perturbed by a rough path transport noise.

带Hölder漂移的随机流及带噪声的输运/连续性方程

Stochastic flows for Hölder drifts and transport/continuity equations with noise

该研究通过Zvonkin型变换证明了带Hölder漂移SDE的微分同胚随机流存在性，建立相关抛物PDE的适定性与最优正则性，进而得到带输运噪声的随机输运与连续性方程的适定性，弥补确定性情形下解可能不唯一的问题。

关于二维不可压缩 Euler 方程的连续模的损失与传播

On the Loss and Propagation of Modulus of Continuity for the Two-Dimensional Incompressible Euler Equations

本文研究了二维不可压缩 Euler 方程中比 Dini 连续更粗糙的涡度连续模的传播问题，证明了一族显式连续模可被传播，并构造了一族不可传播的连续模。

On the Navier-Stokes equation perturbed by rough transport noise

We consider the Navier-Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space-time dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier-Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the sol…

Some new well-posedness results for continuity and transport equations, and applications to the chromatography system

We obtain various new well-posedness results for continuity and transport equations, among them an existence and uniqueness theorem (in the class of strongly continuous solutions) in the case of nearly incompressible vector fields, possibly having a blow-up of the BV norm at the initial time. We apply these results (valid in any space dimension) to the k x k chromatography system of conservation laws and to the k x k Keyfitz and Kranzer system, both in one space dimension.

纳维-斯托克斯方程带分数传输噪声作为多尺度动力学极限

Navier-Stokes with a fractional transport noise as a limit of multi-scale dynamics

本文提出了一种新的粗糙路径解方法，用于研究纳维-斯托克斯方程在多尺度动力学极限下的解结构，并展示了其在非线性随机偏微分方程中的广泛应用。

Quasilinear rough partial differential equations with transport noise

We investigate the Cauchy problem for a quasilinear equation with transport rough input of the form $\mathrm{d} u-\partial_i(a^{ij}(u)\partial_j u)\mathrm{d} t =\mathrm{d} \mathbf{X}_t^i(x)\partial_i u_t,$ $u_0\in L^2$ on the torus $\mathbb T^d$, where $\mathbf{X}$ is two-step enhancement of a family of coefficients $(X^i_t(x))_{i=1,\dots d}$, akin to a geometric rough path with Hölder regularity $α>1/3.$ Using energy estimates, we provide sufficient conditions that guarantee existence in any dimension, and uniqueness in the case when $X$ is divergence-free. We then focus on the one-dimensiona…

On uniqueness for a rough transport-diffusion equation

In this Note, we study a transport-diffusion equation with rough coefficients and we prove that solutions are unique in a low-regularity class.

由带粗糙输运噪声的三维Navier-Stokes方程生成的随机动力系统

Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise

针对带粗糙输运噪声的三维Navier-Stokes系统解不唯一、无普遍半群性质的问题，研究者选取满足半群性质且符合能量耗散最大化准则的解，首次构造出该类非唯一SPDE的可测单值随机动力系统。

Hölder estimates for parabolic operators on domains with rough boundary

We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the domain, including a very weak compatibility condition between the Dirichlet boundary part and its complement, we prove Hölder continuity of the solution in space and time.

高阶无界粗糙驱动项

Higher Order Unbounded Rough Drivers

该研究构建了高阶无界粗糙驱动项的框架，将其应用于含两类不同驱动项的粗糙输运方程，证明了该方程的适定性。

Finite Volume Method for a System of Continuity Equations Driven by Nonlocal Interactions

We present a new finite volume method for computing numerical approximations of a system of nonlocal transport equation modeling interacting species. This method is based on the work [F. Delarue, F. Lagoutire, N. Vauchelet, Convergence analysis of upwind type schemes for the aggregation equation with pointy potential, Ann. Henri. Lebesgue 2019], where the nonlocal continuity equations are 10 treated as conservative transport equations with a nonlocal, nonlinear, rough velocity field. We analyze some properties of the method, and illustrate the results with numerical simulations.

A (rough) pathwise approach to a class of non-linear stochastic partial differential equations

We consider nonlinear parabolic evolution equations of the form $\partial_{t}u=F(t,x,Du,D^{2}u) $, subject to noise of the form $H(x,Du) \circ dB$ where $H$ is linear in $Du$ and $\circ dB$ denotes the Stratonovich differential of a multidimensional Brownian motion. Motivated by the essentially pathwise results of [Lions, P.-L. and Souganidis, P.E.; Fully nonlinear stochastic partial differential equations. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), no. 9] we propose the use of rough path analysis [Lyons, T. J.; Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (…

Global existence for the stochastic Boussinesq equations with transport noise and small rough data

In this paper, we consider the stochastic Boussinesq equations on $\mathbb T^3$ with transport noise and rough initial data. We first prove the existence and uniqueness of the local pathwise solution with initial data in $L^p(Ω;L^p)$ for $p>5$. By assuming additional smallness on the initial data and the noise, we establish the global existence of the pathwise solution.

粗糙密度的最优传输映射的逐点Schauder估计

Pointwise Schauder estimates for optimal transport maps of rough densities

本文研究了粗糙密度下的最优传输问题，在密度仅于某$L^p$意义下接近常数的条件下，证明了最优传输映射势函数的逐点$C^{2,\,α}$估计。

Compressible Navier--Stokes system with transport noise

We consider the barotropic Navier--Stokes system driven by a physically well-motivated transport noise in both continuity as well as momentum equation. We focus on three different situations: (i) the noise is smooth in time and the equations are understood as in the sense of the classical weak deterministic theory, (ii) the noise is rough in time and we interpret the equations in the framework of rough paths with unbounded rough drivers and (iii) we have a Brownian noise of Stratonovich type and study the existence of martingale solutions. The first situation serves as an approximation for (ii…

Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it do…

Unbounded rough drivers

We propose a theory of linear differential equations driven by unbounded operator-valued rough signals. As an application we consider rough linear transport equations and more general linear hyperbolic symmetric systems of equations driven by time-dependent vector fields which are only distributions in the time direction.

度量图与分形上的连续性方程与输运型方程研究

On equations of continuity and transport type on metric graphs and fractals

该研究针对鞅维数1的度量空间上一阶连续性与输运方程弱解不唯一的问题，借助边界四元组与半群理论证明无散场下的适定性，为分形空间标量解一阶方程适定性研究提供了首个结果。

A pathwise stochastic Landau-Lifshitz-Gilbert equation with application to large deviations

Using a rough path formulation, we investigate existence, uniqueness and regularity for the stochastic Landau-Lifshitz-Gilbert equation with Stratonovich noise on the one dimensional torus. As a main result we show the continuity of the so-called Itô-Lyons map in the energy spaces $L^\infty(0,T;H^k)\cap L^2(0,T;H^{k+1})$ for any $k\ge1$. The proof proceeds in two steps. First, based on an energy estimate in the aforementioned space together with a compactness argument we prove existence of a unique solution, implying the continuous dependence in a weaker norm. This is then strengthened in the …

Optimal design problems in rough inhomogeneous media. Existence theory

This paper settles the existence question for a rather general class of convex optimal design problems with a volume constraint. In low dimensions, we prove the existence of an optimal configuration for general convex minimization problems ruled by bounded measurable degenerate elliptic operators. Under a mild continuity assumption on the medium, the free boundary is proven to enjoy the appropriate weak geometry and we establish the existence of an optimal design for general convex optimal design problems with volume constraints for all dimensions.

Continuity equation and vacuum regions in compressible flows

We investigate the creation and properties of eventual vacuum regions in the weak solutions of the continuity equation, in general, and in the weak solutions of compressible Navier--Stokes equations, in particular. The main results are based on the analysis of renormalized solutions to the continuity and pure transport equations and their inter-relations which are of independent interest.

Smooth approximation is not a selection principle for the transport equation with rough vector field

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation.

迎风方案在粗糙系数下的收敛速率

Convergence rates for upwind schemes with rough coefficients

本文研究了在空间Sobolev正则性速度场和初始数据仅为可积的情况下，迎风有限体积显式方案在求解连续方程中的数值分析。通过Kantorovich-Rubinstein距离估计近似解与连续问题分布解之间的误差，并证明了弱收敛率至少为网格尺寸的1/2次方。

黎曼度量下动力最优传输的偏微分方程特性

PDE aspects of the dynamical optimal transport in the Lorentzian setting

本文探讨了在时空中最优传输问题的等价性，推导了洛伦兹版本的Benamou-Brenier公式。

Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data

In this paper we study the semiclassical limit of the Schrödinger equation. Under mild regularity assumptions on the potential $U$ which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness and stability results for the flow in the space of measures induced by the continuity equation.

具有一般传输算子和粗糙系数的动力学 Fokker-Planck 方程的速度平均与 Hölder 正则性

Velocity averaging and Hölder regularity for kinetic Fokker-Planck equations with general transport operators and rough coefficients

本文研究具有一般传输算子和粗糙系数的动力学 Fokker-Planck 方程弱解的局部有界性和 Hölder 连续性，通过结合椭圆 De Giorgi-Nash-Moser 理论与速度平均引理的方法，利用扩散与传输的混合效应及亚椭圆结构证明了该结果。

A Directional Lipschitz Extension Lemma, with Applications to Uniqueness and Lagrangianity for the Continuity Equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assump…

Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and Hölder continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general conditions including quadratic costs and unbounded marginals. The results for the value extend to regularization by an arbitrary divergence. As an application, we show convergence of Sinkhorn's algorithm in Wasserstein sense, including for quadratic cost. Two techniques are presented: The first compares an optimal coupling with its so-called shadow, a coupling in…

Local regularity result for an optimal transportation problem with rough measures in the plane

We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the measure behaves as an absolutely continuous measure up to that scale. Our main theorem then proves that such convex functions cannot exhibit any flat part at a scale larger than the corresponding discrete scales on the measures. This, in turn, implies a $C^1$ regularity result u…

具有粗糙传输噪声的原始方程：整体适定性与正则性

The primitive equations with rough transport noise: Global well-posedness and regularity

本文利用 $q>2$ 的 $L^q$-技术处理粗糙传输噪声，为原始方程识别出新的临界各向异性 Besov 空间，从而建立了其整体适定性与瞬时正则化结果。

Renormalized solutions to the continuity equation with an integrable damping term

We consider the continuity equation with a nonsmooth vector field and a damping term. In their fundamental paper, DiPerna and Lions proved that, when the damping term is bounded in space and time, the equation is well posed in the class of distributional solutions and the solution is transported by suitable characteristics of the vector field. In this paper, we prove existence and uniqueness of renormalized solutions in the case of an integrable damping term, employing a new logarithmic estimate inspired by analogous ideas of Ambrosio, Lecumberry, and Maniglia, Crippa and De Lellis in the Lagr…

Uniqueness and Lagrangianity for solutions with lack of integrability of the continuity equation

We deal with the uniqueness of distributional solutions to the continuity equation with a Sobolev vector field and with the property of being a Lagrangian solution, that means transported by a flow of the associated ordinary differential equation. We work in a framework with lack of local integrability of the solution, in which the classical DiPerna-Lions theory of uniqueness and Lagrangianity of distributional solutions does not apply due to the insufficient integrability of the commutator. We introduce a general principle to prove that a solution is Lagrangian: we rely on a disintegration al…

Continuity of solutions to space-varying pointwise linear elliptic equations

We consider pointwise linear elliptic equations of the form $\mathrm{L}_x u_x = η_x$ on a smooth compact manifold where the operators $\mathrm{L}_x$ are in divergence form with real, bounded, measurable coefficients that vary in the space variable $x$. We establish $\mathrm{L}^{2}$-continuity of the solutions at $x$ whenever the coefficients of $\mathrm{L}_x$ are $\mathrm{L}^{\infty}$-continuous at $x$ and the initial datum is $\mathrm{L}^{2}$-continuous at $x$. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a …

Results on the diffusion equation with rough coefficients

We study the behaviour of the solutions of the stationary diffusion equation as a function of a possibly rough ($L^{\infty}$-) diffusivity. This includes the boundary behaviour of the solution maps, associating to each diffusivity the solution corresponding to some fixed source function, when the diffusivity approaches infinite values in parts of the medium. In $n$-dimensions, $n \geq 1$, by assuming a weak notion of convergence on the set of diffusivities, we prove the strong sequential continuity of the solution maps. In 1-dimension, we prove a stronger result, i.e., the unique extendability…