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We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2

In this paper, we prove the first existence result of weak solutions to the 3D Euler equation with initial vorticity concentrated in a circle and velocity field in $C([0,T],L^{2^-})$. The energy becomes finite and decreasing for positive times, with vorticity concentrated in a ring that thickens and moves in the direction of the symmetry axis. With our approach, there is no need to mollify the initial data or to rescale the time variable. We overcome the singularity of the initial data by applying convex integration within the appropriate time-weighted space.

For any $2

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18,17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method., Comment: 40 pages, 3 figures

We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable H\"older space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a "turbulence" zone which grows linearly in time around the vortex sheet. As a byproduct, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength., Comment: 41 pages, 2 figures

In this work we extend the results in [6,32] on the 2D IPM system with constant viscosity (Atwood number $A_{\mu}=0$) to the case of viscosity jump ($|A_{\mu}|<1$). We prove a h-principle whereby (infinitely many) weak solutions in $C_tL_{w^*}^{\infty}$ are recovered via convex integration whenever a subsolution is provided. As a first example, non-trivial weak solutions with compact support in time are obtained. Secondly, we construct mixing solutions to the unstable Muskat problem with initial flat interface. As a byproduct, we check that the connection, established by Sz\'ekelyhidi for $A_{\mu}=0$, between the subsolution and the Lagrangian relaxed solution of Otto, holds for $|A_{\mu}|<1$ too. For different viscosities, we show how a pinch singularity in the relaxation prevents the two fluids from mixing wherever there is neither Rayleigh-Taylor nor vorticity at the interface., Comment: 38 pages

We prove the existence of infinitely many mixing solutions for the Muskat problem in the fully unstable regime displaying a linearly degraded macroscopic behaviour inside the mixing zone. In fact, we estimate the volume proportion of each fluid in every rectangle of the mixing zone. The proof is a refined version of the convex integration scheme presented in [DS10, Sze12] applied to the subsolution in [CCF16]. More generally, we obtain a quantitative h-principle for a class of evolution equations which shows that, in terms of weak*-continuous quantities, a generic solution in a suitable metric space essentially behaves like the subsolution. This applies of course to linear quantities, and in the case of IPM to the power balance $\mathbf{P}$ (14) which is quadratic. As further applications of such quantitative h-principle we discuss the case of vortex sheet for the incompressible Euler equations., Comment: 28 pages