Weak stability of Lagrangian solutions to the semigeostrophic equations

In [1], Cullen and Feldman proved existence of Lagrangian solutions for the semigeostrophic system in physical variables with initial potential vorticity in $L^p$, $p>1$. Here, we show that a subsequence of the Lagrangian solutions corresponding to a strongly convergent sequence of initial potential vorticities in $L^1$ converges strongly in $L^q$, $q<\infty$, to a Lagrangian solution, in particular extending the existence result of Cullen and Feldman to the case $p=1$. We also present a counterexample for Lagrangian solutions corresponding to a sequence of initial potential vorticities converging in $\mathcal{BM}$. The analytical tools used include techniques from optimal transportation, Ambrosio's results on transport by $BV$ vector fields, and Orlicz spaces. [1] M. Cullen and M. Feldman, {\it Lagrangian solutions of semigeostrophic equations in physical space.} SIAM J. Math. Anal., {\bf 37} (2006), 1371--1395.Comment: 19 page

Existence of Eulerian solutions to the semigeostrophic equations in physical space: the 2-dimensional periodic case

In this paper we use the new regularity and stability estimates for Alexandrov solutions to Monge-Ampere equations estabilished by G.De Philippis and A.Figalli to provide a global in time existence of distributional solutions to a semigeostrophic equation on the 2-dimensional torus, under very mild assumptions on the initial data. A link with Lagrangian solutions is also discussed.Comment: 16 pages, no figure

H\"older regularity of the 2D dual semigeostrophic equations via analysis of linearized Monge-Amp\`ere equations

We obtain the H\"older regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity. Our main tool is an interior H\"older estimate in two dimensions for an inhomogeneous linearized Monge-Amp\`ere equation with right hand side being the divergence of a bounded vector field. As a further application of our H\"older estimate, we prove the H\"older regularity of the polar factorization for time-dependent maps in two dimensions with densities bounded away from zero and infinity. Our applications improve previous work by G. Loeper who considered the cases of densities sufficiently close to a positive constant.Comment: v2: title slight changed; some typos fixe

Solutions of the fully compressible semi-geostrophic system

The fully compressible semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures

Variational Methods for Evolution: Abstracts from the workshop held December 4–10, 2011

The meeting focused on the last advances in the applications of variational methods to evolution problems governed by partial differential equations. The talks covered a broad range of topics, including large deviation and variational principles, rate-independent evolutions and gradient flows, heat flows in metric-measure spaces, propagation of fracture, applications of optimal transport and entropy-entropy dissipation methods, phasetransitions, viscous approximation, and singular-perturbation problems

Equazione del trasporto e problema di Cauchy per campi vettoriali debolmente differenziabili

"Transport equation and Cauchy problem for weakly differentiable vector fields". In this thesis we study the well-posedness of the Cauchy problem and of the transport equation, assuming a very low regularity on the vector field. At first we give an overview of the problem and illustrate the classical setup. Then we summarize the classical results of DiPerna and Lions about vector fields with Sobolev regularity and the recent results of Ambrosio about vector fields with bounded variation, focussing on the notion of renormalized solution. We also describe a recent paper (joint work with Ambrosio and Maniglia) in which we prove the renormalization property for special vector fields with bounded deformation. The proof is obtained studying the fine properties of the normal trace of vector fields with measure divergence, for which we show an important chain-rule formula. In the last part of the thesis we illustrate some counterexamples to the uniqueness for the transport equation and we address some open problems