Nonlinear PDEs & Gradient Flows:
Analytical and Numerical Aspects
TU Wien, November 21, 2016
Abstracts
Nadia Ansini (Sapienza University of Rome, Italy) Gradient flows with wiggly potential: a variational approach to the dynamics Variational techniques and global minimisation have been proven to be very successful in many applications in material science. The notion of Γconvergence has been introduced to study the asymptotic behaviour of (global) minimizers of energy functionals in the limit when the parameters (related to the multiscale nature of the problem) get small. Even if Γconvergence may fail in giving the correct description of the effect of local minimizers, variational techniques can be still applied to follow the pattern of the local minimizers of energy functionals. In this seminar I will present some recent results on microstructure evolution in materials undergoing martensitic phase transition (gradient flows with wiggly potentials). Minimising movements are investigated for an energy which is the superposition of a convex functional and fast small oscillations. Thus a minimising movement scheme involves a temporal parameter τ and a spatial parameter ε, with τ describing the time step and the frequency of the oscillations being proportional to 1/ε. The extreme cases of fast time scales τ<<ε and slow time scales ε<<τ have been investigated in [Braides2014]. In this seminar I will present some recent results on the intermediate (critical) case of finite ratio ε/τ>0. I will show that a pinning threshold exists, with initial data below the threshold being a fixed point of the dynamics. For initial data above the pinning threshold, the equation and velocity describing the homogenised motion are determined. The results are obtained in collaboration with Andrea Braides (Dept. of Mathematics, University of Rome Tor Vergata, Italy) and Johannes Zimmer (Dept. of Mathematical Sciences, University of Bath, UK). References: [Braides2014] A. Braides. Local Minimization, Variational Evolution and Γconvergence. Volume 2094 of Lecture Notes in Mathematics. Springer, 2014. 
John W. Barrett (Imperial College London, UK) Stable parametric finite element approximations for Willmoretype flows We consider parametric finite element approximations for geometric evolution equations arising as gradient flows for energy functionals that are quadratic in the principal curvatures of a twodimensional closed surface. Beside the wellknown Willmore and Helfrich flows, we will also consider flows involving spontaneous curvature and area difference elastic effects, which are important in many applications such as biomembranes. Our approach extends ideas from Dziuk and the present authors to obtain a stable continuousintime semidiscrete approximation that allows for a tangential redistribution of the mesh points, which typically leads to better mesh properties. In addition, we show existence and uniqueness results in the fully discrete case. Finally, we extend the approach to open surfaces with Gaussian curvature effects. This is joint work with Harald Garcke (University of Regensburg) and Robert Nürnberg (Imperial College London). 
Sören Bartels (University of Freiburg, Germany) Gradient flows for constrained geometric partial differential equations The mathematical description of thin elastic objects undergoing large bending deformations leads to energy minimization problems involving a pointwise constraint on the deformation gradient. Corresponding gradient flows may serve as a basis for the construction of numerical methods for minimizing the bending energy or as simple models for the prediction of certain evolution processes. In the case of thin elastic sheets the constraint enforces the deformation to be an isometry so that length and angle relations remain unchanged and that the Gaussian curvature of the deformed sheet vanishes. For thin elastic rods this enforces the object to be inextensible or that the deformation is arclength parametrized. We show that the semiimplicit discretization of the linearized constraint leads to a violation of the constraint that is controlled by the stepsize and still leads to unconditionally stable schemes. The spatial discretization is based on conforming and nonconforming finite element methods for which the deformation gradient is a nodal degree of freedom. 
Bertram Düring (University of Sussex, UK) Lagrangian schemes for Wasserstein gradient flows A wide range of diffusion equations can be interpreted as gradient flow with respect to Wasserstein distance of an energy functional. Examples include the heat equation, the porous medium equation, and the fourthorder DerridaLebowitzSpeerSpohn equation. When it comes to solve equations of gradient flow type numerically, schemes that respect the equation's special structure are of particular interest. The gradient flow structure gives rise to a variational scheme by means of the minimising movement scheme (also called JKO scheme, after the seminal work of Jordan, Kinderlehrer and Otto) which constitutes a timediscrete minimization problem for the energy. While the scheme has been used originally for analytical aspects, a number of authors have explored the numerical potential of this scheme. Such schemes often use a Lagrangian representation where instead of the density, the evolution of a timedependent homeomorphism that describes the spatial redistribution of the density is considered. In this talk we review some results on Lagrangian schemes for Wasserstein gradient flows in one spatial dimension and then discuss extensions to higher approximation order and to higher spatial dimensions. 
Jan Maas (IST Austria, Austria) Gradient flows and entropy inequalities for quantum systems with detailed balance We present a new class of transport metrics for density matrices, which can be viewed as noncommutative analogues of the 2Wasserstein metric. With respect to these metrics, we show that dissipative quantum systems can be formulated as gradient flows for the von Neumann entropy under a detailed balance assumption. We also present geodesic convexity results for the von Neumann entropy in several interesting situations. These results rely on an intertwining approach for the semigroup combined with suitable matrix trace inequalities. This is joint work with Eric Carlen. 
Riccarda Rossi (University of Brescia, Italy) Singular perturbations and balanced viscosity solutions of gradient flows in Hilbert spaces In this talk, based on an ongoing collaboration with Virginia Agostiniani and Giuseppe Savaré, we address the asymptotic behavior as \varepsilon \downarrow 0 of the solutions to the (Cauchy problem for the) gradient flow equation
\varepsilon u'(t) + \mathrm{D} \mathcal{E}(t,u(t)) =0 \qquad \text{in } \mathcal{H}, \ t \in (0,T),
where \mathcal{H} is a separable Hilbert space and \mathcal{E} : (0,T) \times \mathcal{H} \to (\infty,+\infty] a timedependent energy functional, with u \mapsto \mathcal{E}(t,u) possibly nonconvex.
The main difficulty attached to the analysis as \varepsilon \downarrow 0 of a family (u_\varepsilon)_\varepsilon of solutions resides in the lack of estimates on (u_\varepsilon')_\eps.
We develop a variational approach to this singular perturbation problem, based on the study of the limit as \varepsilon \downarrow 0 of the energy identity
\frac{\varepsilon}{2} \int_s^t \vert u_\varepsilon'(r)\vert^2 \mathrm{d}r + \frac{1}{2\varepsilon} \int_s^t \mathrm{D}\mathcal{E}(r,u_\varepsilon(r))^2 \mathrm{d}r + \mathcal{E}(t,u_\varepsilon(t))
\quad \quad = \mathcal{E}(s,u_\varepsilon(s)) + \int_s^t \partial_t \mathcal{E}(r,u_\varepsilon(r)) \mathrm{d}r \quad \text{ for all } 0 \leq s \leq t \leq T,
and in particular on a fine analysis of the asymptotic properties of the quantity
\int_s^t u_\varepsilon'(r) \mathrm{D}\mathcal{E}(r,u_\varepsilon(r)) \mathrm{d}r.
As we will see, in this context there naturally comes into play the condition that for every t\in (0,T) the critical points of \mathcal{E}(t,\cdot) are isolated.
We will discuss the role of this property, its genericity, and hint at a possible weakening of it.

Martin Rumpf (University of Bonn, Germany) A constrained optimization approach for the evolution of thin films on curved surfaces The motion of a thin viscous film of fluid on a curved surface exhibits many intricate visual phenomena, which are challenging to simulate using existing techniques. In the talk a reduced model, involving only the temporal evolution of the mass density of the film on the surface is investigated. However, in this model, the motion is governed by a fourthorder nonlinear PDE, which involves geometric quantities such as the curvature of the underlying surface, and is therefore difficult to discretize. A variational time discretization for this problem is introduced and discretized on triangle meshes. To this end, a discretization for the curvature and advection operators is introduced which leads to an efficient and stable numerical scheme, requires a single sparse linear solve per time step, and exactly preserves the total volume of the fluid. This method will be validated experimentally by a qualitative comparison to known results from the literature. Furthermore, the talk will demonstrate various intricate effects achievable by this method, such as droplet formation, evaporation, droplets interaction and viscous fingering. Finally, the method is extended to incorporate nonlinear van der Waals forcing terms which stabilize the motion of the film and to allow additional effects such as pearling. The talk is based on a cooperation with Omri Azencot, Miri Ben Chen, Orestis Vantzos, and Max Wardetzky. 