## Final Report Summary - GFTIPFD (Geometric function theory, inverse problems and fluid dinamics)

The project strike to conquer novel results in geometric function theory inverse problems and fluid dynamics. In geometric function theory (motivated by the theory of differential inclussions and random surfaces ) we have developed a program to understand the linera nonlinear Beltrami equation. Apart from expected results as Schauder estimates, or non negativity of the Jacobian where the nonlinearity is Hólder continuous we have shown that the set of solutions form a 2 dimensional manifold, and a surprising Higuer Hölder regularity in the automous case. Besides we have shown that the combination of staircase laminates with convex integration provides nonlinearities with low Weyl exponent. A sophisticated variant of this technique yields example of pathological homeomorphism with sharp integrability properties in all dimensions. In our studies of the Calderón problem we understood linear Beltrami equation with Sobolev regularity. Carlos Mora, has developed a solid mapping theory for invertible Sobolev maps compatible with fracture and cavitation.

Part of the strength of the project lies in its interdisciplinary nature. As a matter of fact ideas from

GFT have had an impact in our studies in Inverse Problems. For example Conformal Invariant Tensor give obstructions to those manifolds where the Calderón Problem can be solved with the technique of CGO solutions. A careful analysis of the Weyl and Cotton tensor algebraically provide us with a sufficient condition and also shows that such metrics are rare quantitatively. We have also understood the relation of such good manifolds with generalized surface of revolution or generalized unimodular Lie Groups for counterexamples. Another aspect of this fruitful interaction is seen in the two dimensional Calderón problem.

Here, subtle properties of the Beltrami equation have allowed us to obtain if and only if conditions for the stability of Calderón problem in 2d solving a conjecture of G.Alesssandrini from 2007. Moreover we have quantified very precisely the old idea that oscillating coeficcients can not be recovered in a stable manner, stablishing a precise relation between G-convergence and convergence of Dirichlet to Neumann maps.

Building more in this transversal point of view of these well known problems we have found a surprising connection of Buckgheim method to recover potentials from the D-N map(or the fixed energy scattering data) and the convergence to initial data of the time dependent non elliptic equation. This yield us recovery formulas, which we show with an explicit example that fail with half derivative square integrable. However we have discovered that by controlling the discontinuities or by defining suitable averages this threshold can be bypassed and almost all potentials can be recovered with a robust algorithm.

The ideas of convex integration have recently gained a reboost as they are key to obtain wild solutions in hydrodynamics. One of the main aims of the project was to understand the Muskat Problem in the unstable case. We have found that convex integration and subtle elaboration on contour dynamics yield mixing solutions where the mixing zone is describe as a suitable neighborhood of a pseudointerphase evolving in time. The evolution in time is very subtle and also beautiful as it is needed to understand the govering pseudodifferential operator with the philosophy of semiclassical analysis (that is our PSO behave as classical Fourier multipliers up to factors which are proportional to the time) where the symbols are not smooth. Our mixing zone resembles the fingering phenomena that one sees in the experiments. In this context we have also look at magnetohydronamics. In contrast with other similar equations here there are quantities which are conserved for non-smooth solutions. This has an impact in convex integration as provides nonlinear constraints. We have seen that no convex integration with 0 initial data exists in 2d but it does exist in 3D. Compensated compactness is the right tool to understand magnetic helicity which have allow us to prove a conjecture of Taylor on the conservation of magnetic helicity for weak limits of Leray-Hopff solutions in the vanishing resistivity limit. Thus the project has been very satisfactory and at least 7 postdoctoral researchers and three phd students have been trained.

Part of the strength of the project lies in its interdisciplinary nature. As a matter of fact ideas from

GFT have had an impact in our studies in Inverse Problems. For example Conformal Invariant Tensor give obstructions to those manifolds where the Calderón Problem can be solved with the technique of CGO solutions. A careful analysis of the Weyl and Cotton tensor algebraically provide us with a sufficient condition and also shows that such metrics are rare quantitatively. We have also understood the relation of such good manifolds with generalized surface of revolution or generalized unimodular Lie Groups for counterexamples. Another aspect of this fruitful interaction is seen in the two dimensional Calderón problem.

Here, subtle properties of the Beltrami equation have allowed us to obtain if and only if conditions for the stability of Calderón problem in 2d solving a conjecture of G.Alesssandrini from 2007. Moreover we have quantified very precisely the old idea that oscillating coeficcients can not be recovered in a stable manner, stablishing a precise relation between G-convergence and convergence of Dirichlet to Neumann maps.

Building more in this transversal point of view of these well known problems we have found a surprising connection of Buckgheim method to recover potentials from the D-N map(or the fixed energy scattering data) and the convergence to initial data of the time dependent non elliptic equation. This yield us recovery formulas, which we show with an explicit example that fail with half derivative square integrable. However we have discovered that by controlling the discontinuities or by defining suitable averages this threshold can be bypassed and almost all potentials can be recovered with a robust algorithm.

The ideas of convex integration have recently gained a reboost as they are key to obtain wild solutions in hydrodynamics. One of the main aims of the project was to understand the Muskat Problem in the unstable case. We have found that convex integration and subtle elaboration on contour dynamics yield mixing solutions where the mixing zone is describe as a suitable neighborhood of a pseudointerphase evolving in time. The evolution in time is very subtle and also beautiful as it is needed to understand the govering pseudodifferential operator with the philosophy of semiclassical analysis (that is our PSO behave as classical Fourier multipliers up to factors which are proportional to the time) where the symbols are not smooth. Our mixing zone resembles the fingering phenomena that one sees in the experiments. In this context we have also look at magnetohydronamics. In contrast with other similar equations here there are quantities which are conserved for non-smooth solutions. This has an impact in convex integration as provides nonlinear constraints. We have seen that no convex integration with 0 initial data exists in 2d but it does exist in 3D. Compensated compactness is the right tool to understand magnetic helicity which have allow us to prove a conjecture of Taylor on the conservation of magnetic helicity for weak limits of Leray-Hopff solutions in the vanishing resistivity limit. Thus the project has been very satisfactory and at least 7 postdoctoral researchers and three phd students have been trained.