Partial differential equations on fractals. Existence, Uniqueness and Approximation results
Creators: name: Meinert, Melissa
Publisher: name: Universität Bielefeld
Year: 2020
Subjects: subject: 510

description: In this thesis, we investigate partial differential equations involving first order terms on fractal spaces, and our main interest is to provide graph approximations for their solutions.<br><br>
The first part contains a survey of Dirichlet and resistance forms on certain fractal spaces and we also recall basics on metric graphs. Moreover, we provide basic concepts of the analysis of resistance forms. We close this chapter by presenting some examples of spaces that carry a local regular resistance form in the sense of Kigami.<br><br>
Existence and uniqueness results are presented in the second part.
After a brief discussion of fractal analogs of known existence and uniqueness results for linear elliptic and parabolic partial differential equations of second order, we investigate a nonlinear partial differential equation, namely the viscous Burgers equation. We discuss adequate formulations of the viscous Burgers equation and prove existence, uniqueness and continuous dependence on initial conditions for a vector-valued Burgers equation on metric graphs. We also consider the Burgers equation on compact resistance spaces and again we state existence, uniqueness and continuous dependence on initial conditions. The proofs are minor modifications compared to the metric graph case.
Furthermore, we show existence of weak solutions to first order equations of continuity type associated to suitably defined vector fields. Our proof is based on a classical vanishing viscosity argument. 
Up to this point it is not necessary that the form under consideration admits a carré du champ, so the volume measure can be more general. 
The last part of this chapter concerns _p_-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form having a carré du champ. These Sobolev spaces are then used to generalize some basic results from the calculus of variations, such as the existence of minimizers for convex functionals and certain constrained minimization problems. This applies to a number of non-classical situations such as degenerate diffusions, superpositions of diffusions and diffusions on fractals equipped with a Kusuoka type measure or to products of such fractals.<br><br>
The third part is the heart of the thesis and deals with approximation results. 
We start again with linear elliptic and parabolic partial differential equations on resistance spaces which involve gradient and divergence terms. For equations on a single resistance space but with varying coefficients we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If the coefficients converge, we can conclude the uniform convergence of the solutions. We then consider equations on a sequence of resistance spaces approximating a target resistance space from within. Under suitable assumptions extensions of linearizations of solutions along this sequence accumulate or even converge uniformly to the solution on the target space. Examples include graph approximations for finitely ramified spaces and metric graph approximations for post-critically finite self-similar spaces.
Next, we consider the viscous Burgers equation on a post-critically finite self-similar fractal associated with a regular harmonic structure. Using Post's concept of generalized norm resolvent convergence on varying Hilbert spaces we prove that solutions to the Burgers equation can be approximated in a certain weak sense by solutions to corresponding equations on approximating metric graphs. 
Finally, we also show that a sequence of solutions to the viscous continuity equation on graphs approximating a finitely ramified fractal converges along a subsequence to a solution to the continuity equation, provided that certain assumptions on the vector fields are satisfied. The proof relies on a diagonal compactness argument combining vanishing diffusion together with a convergence scheme on varying Hilbert spaces in the sense of Kuwae and Shioya.