BCAM Course

Introduction to Compressible Navier-Stokes. Derivation of the model via physical laws (Conservation of mass and momentum). Energy inequality for the system. Definition of the weak solutions. Concept of renormalized solutions. Heuristically explain the main idea behind the existence proof: i. compactness of weak solutions, ii. Estimates from energy, iii. Improved estimates of density, iv. Limit passage, v. Effective viscous flux, vi. Strong Convergence of density.

Construction of approximate solutions: three levels of approximations. Boundedness of the density for approximate levels.

Existence of global solution in the Galerkin level. Derive several estimates independent of the dimension of Galerkin approximations.

Existence of complete system with dissipation in the continuity equation and artificial pressure in the momentum equation.

The vanishing viscosity limit. Passing to the limit in the artificial pressure.

Local-in-time existence of strong solutions: i. Lagrangian change of variables, ii. Analysis of linear problem, iii. Estimates of the nonlinear terms, iv. Fixed point argument.

Global-in-time existence with small data: i. A priori estimates, ii. Extension of solution.

Notion of Relative energy. Finite energy weak solutions satisfy the relative energy inequality.

Weak-strong uniqueness on bounded domains.

Currently, the speaker is working in the Basque Center of Mathematics (BCAM) as an Ikerbasque Research Fellow and Ramón y Cajal Fellow. His main area of research is the Fluid flows and Fluid-Structure Interaction (FSI) problems, particularly, existence, uniqueness, long time behaviour and control aspects (controllability, stabilizability and optimal control) of FSI problems. These problems  come from different domains of application, such as medicine (blood motion in arter- ies), biology (fish swimming, bird flight, micro-organism motion), civil engineering  (bridge construction), naval and aerospace engineering etc. Despite all their practical applications, the mathematical understanding of such models is very challenging due to the involvement of strong nonlinearities and the presence of free boundaries.

 The speaker completed his Ph.D from Tata Institute of Fundamental Research, Centre For Applicable Mathematics (TIFR-CAM), Bangalore. He had Postdoctoral research experience in several institutes: INRIA Nancy-Grand Est, France (2019-20), Czech Academy of Science, Prague (2020-2021) and Basque Center of Mathematics (BCAM), Spain (2021-2022) respectively. After that, he was awarded the prestigious Humboldt fellowship  (supported by the Alexander von Humboldt-Stiftung / Foundation) for 2 years (2022-2024) and during this time he was working at Technische Universität Darmstadt.