Analysis and Probability seminar

Abstract: We consider non-linear parabolic equations in a rough setting. For simplicity, we take a non-linear heat equation as an example, which is driven by an elliptic operator with real, bounded, elliptic, measurable coefficients, and whose non-linearity is a (superlinear) power of the solution. The initial value is taken from a homogeneous Besov space (of negative regularity) which is scaling-critical for the equation. The novel aspect in our project is the treatment of rough coefficients with an initial value that does not fit into a variational setting. Rough coefficients are relevant to model, for instance, diffusion processes in compound materials. In my talk, I will discuss two aspects of the project: 1) the natural extension spaces to homogeneous Besov spaces are so-called Z-spaces. They are related to tent spaces, which proved their merit in non-linear PDEs by the celebrated Koch—Tataru theorem for the Navier-Stokes equations. I will explain why I consider Z-spaces as the more suitable solution spaces for most non-linear parabolic PDEs nevertheless. 2) Due to the rough coefficients, elliptic regularity theory is limited. Therefore, all our main results are formulated in the language of weak solutions (in the PDE sense). Nevertheless, also a notion of ‘mild-solution’ is crucial for us, and I will explain the fruitful interaction between the two notions. Indeed, passing from one side to another and vice versa is fundamental to obtain the best conclusion in the end.