Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields

Ambrosio, Luigi; Crippa, Gianluca

doi:10.1007/978-3-540-76781-7_1

Cited by 100 publications

(184 citation statements)

References 78 publications

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“…For this fact the reader might consult one of the references [2], [3] or [11]. Using Theorem 2.3 and the above "backward" ODE, it is easy to see that under the assumptions of Theorem 2.3 we conclude the convergence in measure of the maps X…”

Section: Regular Lagrangian Flowsmentioning

confidence: 93%

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Transport equations with integral terms: existence, uniqueness and stability

2016

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We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna-Lions approach, the essential step is to formulate the problem in the Lagrangian setting. Some motivations to study the above problem arise from the description of polymeric flows, where such kind of equations are coupled with other Navier-Stokes type equations. Using the results for the transport equation we will provide, in a separate paper, a sequential stability theorem for the full problem of the flow of concentrated polymers. At the end of the note we also point out a relevant example about the strong stability of the continuity equation, which highlights the role of an important assumption in our main stability statement. TRANSPORT EQUATION WITH INTEGRAL TERMSCAMILLO DE LELLIS, PIOTR GWIAZDA, AND AGNIESZKAŚWIERCZEWSKA-GWIAZDA Abstract. We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna-Lions approach, the essential step is to formulate the problem in the Lagrangian setting. Some motivations to study the above problem arise from the description of polymeric flows, where such kind of equations are coupled with other Navier-Stokes type equations. Using the results for the transport equation we will provide, in a separate paper, a sequential stability theorem for the full problem of the flow of concentrated polymers.

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Section: Regular Lagrangian Flowsmentioning

confidence: 93%

“…Indeed the DiPerna-Lions theory ensures that the solutions to the continuity equations are stable under the convergence in Theorem 2.3, cf. [3]. Note, moreover, that we have the renormalized property for solutions of the continuity equation in the following form: [3,Theorem 24]).…”

Section: (240)mentioning

confidence: 99%

Transport equations with integral terms: existence, uniqueness and stability

2016

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“…For simplicity, we state it only in the case of equibounded vector fields (see [11] for more general results).…”

Section: Vector Fields With Sobolev Spatial Regularitymentioning

confidence: 99%

“…x of the flow X(t, x) (see [11] for a more detailed treatment of this topic). These results provide a sort of bridge to the standard Cauchy-Lipschitz calculus.…”

Section: Theorem 410 (Stability) Letmentioning

confidence: 99%

Continuity equations and ODE flows with non-smooth velocity

Ambrosio¹,

Crippa²

2014

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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119

124

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In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable 'weak differentiability' assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

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“…As is known [2], under Assumption (A0) there exists a unique distributional solution to (4). Moreover, this solution can be written in the form…”

Section: Continuity Equationmentioning

confidence: 99%

Optimal control of continuity equations

Pogodaev

2016

Nonlinear Differ. Equ. Appl.

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An optimal control problem for the continuity equation is considered. The aim of a "controller" is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.2000 Mathematics Subject Classification: 49K20, 49J15

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Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields

Cited by 100 publications

References 78 publications

Transport equations with integral terms: existence, uniqueness and stability

Transport equations with integral terms: existence, uniqueness and stability

Continuity equations and ODE flows with non-smooth velocity

Optimal control of continuity equations

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