6 research outputs found
Logarithmic estimates for continuity equations
The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in can be used to provide an alternative proof of the result in, where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions
Positive solutions of transport equations and classical nonuniqueness of characteristic curves
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 then provide
a negative answer using tools introduced by Modena and Sz\'ekelyhidi in the
recent groundbreaking work [Ann. PDE, 4, 2018]. On the opposite side, we
introduce a new class of asymmetric Lusin-Lipschitz inequalities and use them
to prove the uniqueness of positive solutions of the continuity equation in an
integrability range which goes beyond the DiPerna-Lions theory
Classical flows of vector fields with exponential or sub-exponential summability
We show that vector fields whose spatial derivative satisfies a
Orlicz summability condition have a spatially continuous representative and are
well-posed. For the case of sub-exponential summability, their flows satisfy a
Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if
satisfies a suitable exponential summability condition then the flow
associated to has Sobolev regularity, without assuming boundedness of . We then apply these results to the representation and Sobolev
regularity of weak solutions of the Cauchy problem for the transport and
continuity equations.Comment: 35 page
Logarithmic estimates for continuity equations
The aim of this short note is twofold. First, we give a sketch of the proof of a recent result proved by the authors in the paper [7] concerning existence and uniqueness of renormalized solutions of continuity equations with unbounded damping coefficient. Second, we show how the ideas in [7] can be used to provide an alternative proof of the result in [6, 9, 12], where the usual requirement of boundedness of the divergence of the vector field has been relaxed to various settings of exponentially integrable functions