On smooth approximations of rough vector fields and the selection of flows

In this work we deal with the selection problem of flows of an irregular vector field. We first summarize an example from \cite{CCS} of a vector field $b$ and a smooth approximation $b_\epsilon$ for which the sequence $X^\epsilon$ of flows of $b_\epsilon$ has subsequences converging to different flows of the limit vector field $b$. Furthermore, we give some heuristic ideas on the selection of a subclass of flows in our specific case.Comment: Proceeding of the "XVII International Conference on Hyperbolic Problems: Theory, Numerics, Applications.

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.Comment: 22 pages, 4 figure

Eulerian and Lagrangian solutions to the continuity and Euler equations with L1L^1 vorticity

In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in \cite{BouchutCrippa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in \cite{Seis16a} and some tools from harmonic analysis introduced in \cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in \cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in \cite{CrippaSpirito15}

Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy

We discuss the Lagrangian property and the conservation of the kinetic energy for solutions of the 2D incompressible Euler equations. Existence of Lagrangian solutions is known when the initial vorticity is in $L^p$ with $1\leq p\leq \infty$. Moreover, if $p\geq 3/2$ all weak solutions are conservative. In this work we prove that solutions obtained via the vortex method are Lagrangian, and that they are conservative if $p>1$.Comment: 28 page

Energy conservation for 2D Euler with vorticity in L(log⁡L)αL(\log L)^\alpha

In these notes we discuss the conservation of the energy for weak solutions of the two-dimensional incompressible Euler equations. Weak solutions with vorticity in $L^\infty_t L^p_x$ with $p\geq 3/2$ are always conservative, while for less integrable vorticity the conservation of the energy may depend on the approximation method used to construct the solution. Here we prove that the canonical approximations introduced by DiPerna and Majda provide conservative solutions when the initial vorticity is in the class $L(\log L)^\alpha$ with $\alpha>1/2$.Comment: arXiv admin note: text overlap with arXiv:1905.0972

Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit

In this paper we prove the uniform-in-time $L^p$ convergence in the inviscid limit of a family $\omega^\nu$ of solutions of the $2D$ Navier-Stokes equations towards a renormalized/Lagrangian solution $\omega$ of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $\omega^\nu$ to $\omega$ in $L^p$. Finally, we show that solutions of the Euler equations with $L^p$ vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach

Advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity

In the first part of the paper, we study the Cauchy problem for the advection-diffusion equation $\partial_t v + \text{div }(v\boldsymbol{b} ) = \Delta v$ associated with a merely integrable, divergence-free vector field $\boldsymbol{b}$ defined on the torus. We establish existence, regularity and uniqueness results for various notions of solutions, in different regimes of integrability both for the vector field and for the initial datum. In the second part of the paper, we use the advection-diffusion equation to build a vanishing viscosity scheme for the transport/continuity equation drifted by $\boldsymbol{b}$, i.e. $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$. Under Sobolev assumptions on $\boldsymbol{b}$, we give two independent proofs of the convergence of such scheme to the Lagrangian solution of the transport equation. One of the proofs is quantitative and yields rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration schemes, as shown in recent works [10, 31, 32, 33], and rules out the possibility of anomalous dissipation.Comment: 28 pages, 2 figure