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Compact course on topics in Viscous Flows

10th - 17th July 2017

About Instructor

Šárka Nečasová is Head of the Department of Evolution DEs and Researcher in the Institute of Mathematics, Czech Academy of Sciences of the Czech Rep., Prague. She is a renowned expert in the mathematical theory of compressible fluid dynamics and fluid-structure interaction problems. She is a recipient of the Wichterle Prize (2003) awarded by the Academy of Science of the Czech Republic for young researches. She is the author/co- authors of more than 129 papers and several larger synthetic works (chapters in encyclopaedias) and one book.

Website: http://www.math.cas.cz/homepage/main_page.php?id_membre=22

Topics of lectures

  1. Viscous compressible Navier-Stokes-(Fourier) system coupled to the radiative transfer equation

  2. Motion of viscous compressible fluids in time dependent domains

  3. Motion of rigid body in viscous fluid

Who can apply ?

The course is aimed towards PhD students, postdocs and faculty who are working in the areas of PDE and its applications in fluid dynamics. The course will be at an advanced theoretical level and the participants will be expected to have basic knowledge in Sobolev Spaces, Functional Analysis and PDE.

Support

A limited number of selected participants will be paid train fare and boarding/lodging during the period of the course.

How to apply ?

Please send a detailed CV which explains your research areas and how it is relevant to this compact course. In addition, if you are a student, please ask your advisor to send a recommendation letter to airbus-chair@tifrbng.res.in before the deadline.

Important Dates

  • Deadline for application : 30th April 2017
  • Annoucement of results : 2nd May 2017

 

Supported by

1. Airbus Chair on Mathematics of Complex Systems, TIFR-CAM, Bangalore.

2. TIFR Centre for Applicable Mathematics, Bangalore.


Course Contents

Viscous compressible Navier-Stokes-(Fourier) system coupled to the radiative transfer equation

We consider relativistic and ”semi-relativistic” models of radiative viscous com- pressible Navier-Stokes-(Fourier) system coupled to the radiative transfer equation extending the classical model introduced in [2]. We concentrate on the problem of existence of weak solution of the problem. Secondly we will study some of its singular limits (low Mach and di↵usion) in the case of well-prepared initial data and Dirichlet boundary condition for the velocity field. In the low Mach number case we prove the convergence toward the incompressible Navier-Stokes system coupled to a system of two stationary transport equations see [3]. Moreover, in the di↵usion case we prove the convergence toward the compressible Navier-Stokes with modified state functions (equilibrium case) or toward the compressible Navier-Stokes coupled to a di↵usion equation (non equilibrium case), see [4, 5].

Motion of viscous compressible fluids in time dependent domains

Second subject we consider the problem of the motion of compressible fluids in domain with varying boundary. We focus on the existence of weak solution and the singular limit in the low Mach number regime, [6, 7].

Motion of rigid body in viscous fluid

We shall consider the problem of the motion of a rigid body in an incompressible viscous fluid filling a bounded domain.This problem was studied by several authors. They mostly considered classical non-slip boundary conditions, which gave them very paradoxical result of no collisions of the body with the boundary of the domain. Only recently there are results when the Navier type of boundary are considered.

In our lecture we shall consider the Navier condition on the boundary of the body and the non-slip condition on the boundary of the domain. This case admits collisions of the body with the boundary of the domain. We shall prove the global existence of weak solution of the problem,[1]


References

  1. [1]  N. Chemetov, Sˇ. Neˇcasova ́, The motion of the rigid body in the viscous fluid including collisions. Global solvability result. Nonlinear Anal. Real World Appl. 34 (2017), 416–445.

  2. [2]  B. Ducomet, E. Feireisl, Sˇ. Neˇcasov ́a: On a model of radiation hydrodynamics. Ann. I. H. Poincar ́e-AN 28 (2011) 797–812.

  3. [3]  B. Ducomet, Sˇ. Neˇcasov ́a: Low Mach number limit in a model of radiative flow, Journal of Evolution equations, 14 (2014) 357–385.

  4. [4]  B. Ducomet, Sˇ. Neˇcasov ́a: Di↵usion limits in a model of radiative flow, Ann. Univ. Ferrara Sez. VII Sci. Mat. 61 (2015), no. 1, 17–59.

  5. [5]  B. Ducomet, Sˇ. Neˇcasova ́: Singular limits in a model of radiative flow, J. Math. Fluid Mech. 17 (2015), 2, 341–380.

  6. [6]  E. Feireisl, O. Kreml, Sˇ. Neˇcasov ́a, J. Neustupa, J. Stebel: Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains. J. Di↵erential Equations 254 (2013), no. 1, 125–140.

  7. [7]  E. Feireisl, O. Kreml, Sˇ. Neˇcasova ́, J. Neustupa, J. Stebel: Incompressible limits of fluids excited by moving boundaries. SSIAM J. Math. Anal. 46 (2014), no. 2, 1456–1471.