Transport equations with integral terms: existence, uniqueness and stability

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. 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.

Optimal stability estimates for continuity equations

“…We include these results in order to demonstrate the strength of the estimates from [38], and to underline the intrinsic connection between the respective works (in particular [37,38]). The first example partly extends the recent analysis in [15].…”

“…This estimate is sharp, as can be seen from the following example, suggested by De Lellis et al . [15]. 1).…”

Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves

“…uniqueness result for trajectories mentioned above. The proofs of the latter and of Theorem 1.5 employ all some suitable Lusin-Lipschitz type estimates for u, an idea pioneered in [4] and [14] and which has proved quite fruitful in different contexts (see for instance [5][6][7]13,16]). As it is well known, for sufficiently regular domains ⊂ R d and when r ∈ (1, ∞], a Borel map u belongs to W 1,r ( , R) if and only if there is a function g ∈ L r ( ) such that…”