On flows associated to Sobolev vector fields in Wiener spaces: an approach à la DiPerna-Lions
Luigi Ambrosio, Alessio Figalli
本文将Sobolev向量场常微分方程的DiPerna-Lions理论推广到抽象Wiener空间的情形。
In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.
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The article studies the transport equation that governes the motion of a fluid in a bounded domain, under the hypothesis of zero velocity at the boundary and supposing the incompressible nature of the fluid. Together with existence and uniqueness results, we study the DiPerna-Lions stability problem, in the case of a bounded domain.
These are notes of a seminar held at the Institute for Problems in Mechanics, RAS in 2003 and aimed at presentation of [R.J. DiPerna and P.L. Lions, Invent. math. 98, 511 (1989)]. We discuss the notion of a generalized solution to a singular Ordinary Differential Equation introduced by DiPerna and Lions. We stress importance of singular dynamic systems from the "philosophia naturalis" point of view, and extend and simplify the original approach by R.J. DiPerna and P.L. Lions. Further extensions are discussed.
We show the existence and uniqueness of a DiPerna-Lions flow for relativistic particles subject to a Lorentz force in an electromagnetic field. The electric and magnetic fields solves the linear Maxwell system in the void but for singular initial conditions. As the corresponding force field is only in $L^2$, we have to perform a careful analysis of the cancellations over a trajectory.
Osgood meets Ambrosio-DiPerna-Lions
本文将 Ambrosio-DiPerna-Lions 理论推广至向量场满足 $L^p$ Osgood 条件的输运方程，通过构造加权能量而非对易子消失的方法，证明其有界分布解唯一且可重正化。
Lyapunov exponents, entropy and mixing for DiPerna-Lions flows
本研究针对不可压缩DiPerna-Lions流构建遍历理论框架，通过Oseledets型分解定义李雅普诺夫指数并结合Ruelle不等式关联度量熵，得到渐近正则性传播与混合速率的精确界，建立Bressan混合猜想的渐近形式。
These notes closely correspond to a series of lectures given by the first author in Toulouse, on the recent extension of the theory of ODE well-posedness to abstract spaces, jointly obtained by the two authors. In the last part, we describe some further developments with respect to the theory of (possibly degenerate) diffusion processes, in a similar setting, contained in the second author's PhD thesis.
Co-moving volumes and Reynolds transport theorem in DiPerna-Lions theory
本文研究共动体积与雷诺输运定理在迪佩拉-洛恩理论中的应用，提出通过修剪可测集来保证测度一致性，并证明了广义流映射下的雷诺输运定理。
Zero-noise selection and Large Deviations in $L^\infty_t L^p_x$ for the stochastic transport equation beyond DiPerna-Lions
本文研究随机输运方程在$L^\infty_t L^p_x$中的解，证明在参数范围内强存在性与路径唯一性，并建立零噪声极限下收敛到DiPerna-Lions重整化解的大偏差原理。
In this paper we provide a complete analogy between the Cauchy-Lipschitz and the DiPerna-Lions theories for ODE's, by developing a local version of the DiPerna-Lions theory. More precisely, we prove existence and uniqueness of a maximal regular flow for the DiPerna-Lions theory using only local regularity and summability assumptions on the vector field, in analogy with the classical theory, which uses only local regularity assumptions. We also study the behaviour of the ODE trajectories before the maximal existence time. Unlike the Cauchy-Lipschitz theory, this behaviour crucially depends on t…
Quantitative Osgood regularity for DiPerna--Lions flows
本文研究DiPerna–Lions流的空间正则性，在端点p=1时证明其满足定量Osgood型估计，方法基于Karamata慢变函数的加权极大算子，结果可应用于输运方程并得到相关正则性与混合尺度下界。
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.
Global renormalized solutions to Boltzmann systems modeling mixture gases of monatomic and polyatomic species
本文研究了Boltzmann系统中混合气体的大型数据初值问题的规范化解，通过构建光滑近似问题并运用平均速度引理，证明了弱$ L^1 $极限为所需解，并验证了解满足熵不等式。
We use some new nonlinear estimates found in \cite {LM} to improve the results of \cite{GL} that establish the acoustic limit for DiPerna-Lions solutions of Boltzmann equation in three ways. First, we enlarge the class of collision kernels treated to that found in \cite{LM}, thereby treating all classical collision kernels to which the DiPerna-Lions theory applies. Second, we improve the scaling of the kinetic density fluctuations with Knudsen number from $O(ε^m)$ for some $m>\frac12$ to $O(ε^\frac12)$. Third, we extend the results from periodic domains to bounded domains with impermeable boun…
A Unified Approach to Mixing and Regularity for Passive Scalar Transport by Sobolev Vector Fields
该研究为索伯列夫向量场的被动标量输运定量估计构建新框架，基于克里斯 - 茹尔内奇异积分估计，得出稳定性估计、指数混合界等成果，涵盖解对初始数据频率依赖、正则性传播及收敛率等多方面结论。
The seminal work of DiPerna and Lions [Invent. Math., 98, 1989] guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio's superposition principle we relate the latter to the uniqueness of positive solutions of the continuity equation and we th…
In the class of Sobolev vector fields in $\mathbb{R}^n$ of bounded divergence, for which the theory of DiPerna and Lions provides a well defined notion of flow, we characterize the vector fields whose flow commute in terms of the Lie bracket and of a regularity condition on the flows themselves. This extends a classical result of Frobenius in the smooth setting.
Recent results on the fluid dynamic limits of the Boltzmann equation based on the DiPerna-Lions theory of renormalized solutions are reviewed in this paper, with an emphasis on regimes where the velocity field behaves to leading order like that of an incompressible fluid with constant density.
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…
A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which the classical DiPerna-Lions-Ambrosio theory of uniqueness of distributional solutions does not apply. We solve partially the open problem that is the case when the vector-field has random dependence. In addition, we prove a stability result for the solutions.
Transport and flow for horizontal Sobolev contact velocities in Carnot groups
本文在Carnot群上建立了由水平Sobolev接触速度场驱动的输运与流方程的适定性，推广了先前仅适用于Heisenberg群的结果，证明基于重正规化解理论。
Nonuniqueness of trajectories on a set of full measure for Sobolev vector fields
本文构造了一个属于W^{1,p}（p<d）的无散度向量场，其流映射在有限时间内将整个环面映射到低维Cantor集，从而证明轨道在全测度集合上非唯一，解决了Alberti的长期问题。
We consider the linear transport equations driven by an incompressible flow in dimensions $d\geq 3$. For divergence-free vector fields $u \in L^1_t W^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the uniqueness of the weak solution in the class $L^\infty_t L^p$ when $\frac{1}{p} + \frac{1}{q} \leq 1$. For such vector fields, we show that in the regime $\frac{1}{p} + \frac{1}{q} > 1$, weak solutions are not unique in the class $ L^1_t L^p$. One crucial ingredient in the proof is the use of both temporal intermittency and oscillation in the convex integrat…
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。
We consider transport equations with an incompressible transporting vector field. Whereas smooth solutions of such equations conserve every $L^p$ norm simply by the chain rule, the question arises how regular a weak solution needs to be to guarantee this conservation property. Whereas the classical DiPerna-Lions theory gives sufficient conditions in terms of the regularity of the coefficients, with no regularity requirement on the transported scalar, we give here sufficient conditions in terms of the combined regularities of the coefficients and the scalar. This is motivated by the case of act…
We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak solutions in the physical energy space. Our method relies on a new $\textit{a priori}$ estimate on the space-time norm in $L^{\f32}_{loc}$ of the Cauchy-Green strain tensor $τ=\F\F^\top$, or equivalently the $L^3_{loc}$ norm of the Jacobian of the flow map $\F$. It allows us to rule out possible concentrations of the energy due to deformations associated with the flow maps. Following the general compactness arguments d…
We prove uniqueness for continuity equations in Hilbert spaces $H$. The corresponding drift $F$ is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna-Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio-Figalli and Fang-Luo, we use a different app…
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…
We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into $\mathbb{R}^\infty$. When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the …
A priori error estimates for the $θ$-method for the flow of nonsmooth velocity fields
本文研究非光滑速度场流动的θ-方法，建立先验误差估计，证明数值解以对数速率收敛于精确解，并通过数值实验验证结果。
Sharp Nonuniqueness in the Transport Equation with Sobolev Velocity Field
本文通过构造反例证明DiPerna--Lions理论中输运方程弱解唯一性条件在Sobolev速度场下的改进范围是尖锐的，并提出一种兼具凸积分与自相似特征的新颖流动机制。
Non-branching properties of Regular Lagrangian Flows and well posedness of the continuity equation in the class of nonnegative solutions
本文证明正则拉格朗日流的特征线分解技术是适定性的充要条件，强化了DiPerna-Lions理论的唯一性结论，并在更弱正则性假设下得到非负解类中连续性方程的适定性结果。
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