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The Muskat problem, which models the flow of immiscible viscous fluids in a porous medium, has been studied extensively in recent years from a number of perspectives. While most studies have focused on harmonic analysis techniques in the graphical setting, the formation of topological singularities makes it desirable to have a notion of weak solution that exists globally in time. In the case of the Muskat problem, this is possible due to the gradient flow structure of the problem relative to the quadratic Wasserstein distance from optimal transport theory, which makes available the JKO or Minimizing Movements scheme . This scheme constructs discrete-in-time solutions for a given timestep $h>0$, then finds a continuous-time solution by taking the limit as $h\to0$. In this dissertation, we study the corresponding weak solutions in two settings.In the first chapter, we study the discrete (in time) JKO solutions to the Muskat problem in a two-dimensional square domain. We show that if the energy of the initial configuration is sufficiently small, then asymptotically any discrete solution is close in $C^1$ to the global equilibrium consisting of a flat interface separating the heavier fluid on bottom from the lighter on top, modulo possible ``drops''. Further, if the surface tension is sufficiently strong, and the discrete solution converges to the global equilibrium, then this convergence occurs exponentially fast.The second chapter uses a modified JKO scheme to model a situation in which a source and sink are present. Using estimates similar to those from the standard scheme, we show convergence of the discrete solutions to a continuous solution, though with some caveats.

We show that the discrete approximate volume preserving mean curvature flow in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set $E$ of the perimeter converges in the long time to a translate of $E$ exponentially fast. As an intermediate result we establish a new quantitative estimate of Alexandrov type for periodic strictly stable constant mean curvature hypersurfaces. Finally, in the two dimensional case a complete characterization of the long time behaviour of the discrete flow with arbitrary initial sets of finite perimeter is provided., 38 pages, 2 figure. arXiv admin note: text overlap with arXiv:2004.04799 by other authors

We consider a class of doubly nonlinear constrained evolution equations which may be viewed as a nonlinear extension of the growing sandpile model of [L. Prigozhin, Eur. J. Appl. Math. 7 (1996) 225–235.]. We prove existence of weak solutions for quite irregular sources by a semi-implicit scheme in the spirit of the seminal works of [R. Jordan et al., SIAM J. Math. Anal. 29 (1998) 1–17, D. Kinderlehrer and N.J. Walkington, Math. Model. Numer. Anal. 33 (1999) 837–852.] but with the 1-Wasserstein distance instead of the quadratic one. We also prove an L1-contraction result when the source is L1 and deduce uniqueness and stability in this case.

We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $\tau$. When the ratio $\frac{\varepsilon}{\tau}$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio between the scales, which is of a different kind. In this case, a discrete interfacial energy is optimized at jump times.

We use the minimizing movement theory to study the gradient flow associated with a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, one can define a Willmore energy on regular functions of L 1 (R d). This functional is extended to every L 1 function by taking its lower semi-continuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions, i.e. functions with balls as level sets. In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then we show that the flow consists on an erosion of the initial data. The erosion speed is given by a first order ordinary equation.

International audience; In Cardaliaguet-Ley (2006) we have defined a viscosity solution for the gradient flow of the exterior Bernoulli free boundary problem. We prove here that the associated energy is non decreasing along the flow. This justifies the "gradient flow" approach for such kind of problem. The proof relies on the construction of a discrete gradient flow in the flavour of Almgren-Taylor-Wang (1993) and on proving it converges to the viscosity solution.