(ANAP) A study of local and nonlocal conservation laws

(ANAP) A study of local and nonlocal conservation laws

Laboratório 4, DMat, UMinho, pólo de Gualtar

2024-05-29 - 14:30

Orador: Eduardo Cardoso de Abreu (UNICAMP, Brasil)

Título: A study of local and nonlocal conservation laws

Data: quarta-feira, 29 de Maio de 2024, 14h30min

Local: Laboratório 4, DMat, UMinho, pólo de Gualtar

Grupo: Análise e Aplicações

Resumo: The interplay between local and nonlocal models of conservation laws have attracted much interest in the past few years motivated by either the need of novel analysis and techniques or by demand to applied sciences. From the existing literature, one can verify that nonlocality in differential problems has been introduced in several ways, and with distinct objectives in mind, for instance, for the modeling hyperbolic-type convective-transport equations for flow motion in porous media, and possibly with the presence of (nonlocal) diffusive term. It is worth noting nonlocal models were introduced also to account for nonlocal interactions over a finite horizon for modeling crowd dynamics and also to improve comprehension on existing models of the vehicular traffic flow dynamics. Nonlocal models appear in fluid mechanics and geophysical problems and related flows given by nonlocal balance laws and fractional diffusion models, for instance,  porous medium equation, diffusion-reaction and surface quasi-geostrophic equation, just to name a few important problems. It is worth noticing the blow-up of solutions linked to local and nonlocal models might lead to remarkable mathematical phenomena either of solutions leading to finite time wave break-down or solutions that can blow-up by forming singularity for bounded data involving a behavior of mass concentration which are distinct and relevant. We will discuss how new insights on the improved concept of no-flow curves/surfaces, in a Lagragian-Eulerian framework, are the key ingredients for numerical approximation and mathematical study of nonlinear hyperbolic problems in classical fluid mechanics and dynamics of fluid in porous media. We will also provide numerical 1D/Multi-D examples to verify the theory and discuss/illustrate the capabilities of the proposed approach.

(Este seminário será seguido de outro após uma pequena pausa.)