# A class of gradient flows of differential forms in negative homogeneous Sobolev spaces

• Eine Klasse von Gradientenflüssen von Differentialformen in negativen homogenen Sobolevräumen

Doemeland, Marco; Melcher, Christof (Thesis advisor); Westdickenberg, Michael (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

In this thesis we are interested in solving a class of quasilinear parabolic partial differential equations (PDEs) for closed differential forms which exhibit a special structure, namely a gradient flow structure, and thus bringing together two major areas of mathematical analysis, the geometric theory of differential forms and the theory of gradient flows. More precisely, for a bounded domain $\Omega \subset \mathbb{R}^n$, $n \ge 2$, with smooth boundary and a time-dependent differential $k$-form $\omega (t) \colon \Omega \rightarrow \Lambda ^k ( \mathbb{R}^n )$ we consider the gradient flow equation $\partial _t \omega \,=\, -\mathop{}\!\mathrm{d} \Big( \nabla c^ \ast \Big[ \mathop{}\!\mathrm{d}^{\ast} \big( \nabla _ \xi F( \, \cdot \, , \omega )\big) \Big] \Big)$ und $\mathop{}\!\mathrm{d} \omega \,=\, 0$. Here, $c \colon \Lambda ^ {k-1}( \mathbb{R}^n ) \rightarrow [0, \infty )$, called the dissipation potential, is a convex function with Legendre-Fenchel dual $c^ \ast$, whereas $\nabla _ \xi F$ denotes the derivative of the energy density $F \colon \Omega \times \Lambda ^k( \mathbb{R}^n ) \rightarrow \mathbb{R}$ with respect to its second argument. This class of PDEs was suggested by Yann Brenier in 2014 as a general framework for dissipative equations and contains for example the $p$-Hodge Laplace heat equation for closed differential forms $\partial _t \omega = - \mathop{}\!\mathrm{d} ( | \mathop{}\!\mathrm{d}^{\ast} \omega | ^ {p-2} \, \mathop{}\!\mathrm{d}^{\ast} \omega )$. The problem of finding weak solutions of the gradient flow equation is challenging not only because of its nonlinearity, but also because of its vectorial character, i.e. it is a system of scalar PDEs. The gradient flow structure appears in the form of a so-called Energy Dissipation Inequality (EDI). Although the latter is equivalent to the gradient flow equation only on a formal level, it nevertheless plays an essential role in establishing the proof of the existence of weak solutions for the PDE. In order the prove the existence of solutions of the corresponding EDI, we use a so-called minimizing movement scheme. This is a time-discrete approximation scheme in which each time-step consists of solving a variational problem. The variational problem involves the perturbation of the energy functional with the so-called dissipation functional which is defined using the dissipation potential $c$. This dissipation functional is closely related to the norms of the duals of homogeneous Sobolev spaces for differential forms, i.e. negative homogeneous Sobolev spaces, which are introduced here. Hence, these spaces define the natural functional analytic setting for the problems addressed in this thesis. To the best of our knowledge, this concept of negative homogeneous Sobolev spaces for differential forms is new. In the limit where the time discretization parameter, used to define the perturbed energy functional, tends to zero the approximation scheme weakly converges to some limit. Since the EDI has well-suited lower semicontinuity properties with respect to the weak convergence, the limit is indeed a solution of the EDI. The limiting process also benefits from compensated compactness methods such as the Sobolev-Poincaré inequality in combination with a Minty-Browder-type argument. With some extra effort we can also prove a reversed EDI for the limit. As the main result of this thesis we conclude from this the existence of a weak solution of the gradient flow equation. In the second part of the thesis we ask for additional properties of the weak solutions of the gradient flow equation such as uniqueness, a semigroup property of the time evolution, an exponential formula as well as error estimates. These problems are very difficult to solve. Because the concept of the EDI is too weak for these questions, we invoke the stronger concept of the Evolution Variational Inequality (EVI). The latter is formally equivalent to the gradient flow equation as well. However, it is only available in the case $c( \xi ) = \frac{1}{2}| \xi | ^2$ in which the dissipation functional becomes, up to a scalar multiple, the negative homogeneous Sobolev norm for the case of Hilbert spaces. As the main results for the second part we prove uniqueness of the limit found before in a class of admissible solutions of the EVI, a contraction property and a semigroup property of the time evolution as well as an exponential formula together with an error estimate. The proof of the exponential formula and the error estimate is given by using two different approaches.