arXiv subjects

Gennaro Ciampa

Publications and source records attributed to Gennaro Ciampa.

21 records · Page 2Linked to original sources

arXiv2019

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.

math.AP↗

arXiv2019

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 $p>1$.

math.AP↗

arXiv2019

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 $b_ε$ for which the sequence $X^ε$ of flows of $b_ε$ 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.

math.AP↗