Resumen de Equacions del transport via nuclis suaus
Juan Carlos Cantero Guardeño
In this thesis we study the non-linear non-local transport equation. This partial differential equation describes the evolution of a scalar which is transported by a velocity field. The field and the scalar are related through the convolution of the latter with a chosen kernel. The main contribution is that, in general, we allow this choice to yield a velocity field having non-zero divergence (unlike the case of the Euler equation) and even non-bounded divergence (unlike the case of the aggregation equation).
The thesis is divided into three blocks.
In the first block, which contains chapters 1 and 2, we study the well-posedness of the equation for a Hölder continuous and compactly supporter initial data. Thus, in chapter 1 we consider a family of kernels in the n-dimensional Euclidean space where each component is a linear combination of derivatives of the fundamental solution of the laplacian. In chapter 2 we work in the complex plane, which allows us to consider an even more general family of kernels.
The second block is the central part of the thesis. We study the problem of density patches, i.e. when the scalar under consideration is the characteristic function of a domain. We recover the result of conservation of the regularity of the boundary of a domain studied by Chemin and Constantin-Bertozzi in 1993 for the Euler equation and also the same result of Bertozzi-Garnett-Laurent-Verdera in 2016 for the aggregation equation. We achieve here a generalisation of these two results for the same families of kernels of the first block: in chapter 3 we do it in arbitrary dimension and in chapter 4 in the complex plane.
The third block is the shortest and corresponds to the last chapter of the thesis. We study the limit behaviour (for infinite time) of a one-dimensional density patch, in this case for the aggregation equation, which is transformed into a transport equation by means of an appropriate change of variables.
