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= Compact Course on Mathematical Aspects of Euler Equations =  = Compact Course on Mathematical aspects of Stochastic Compressible Fluid Flows = 
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Dates : January 21  February 01, 2019  Dates : February 17  February 28, 2020 
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{{attachment:feireisl.jpg align="right",width="400"}}  {{attachment:MartinaHofmanova.jpeg align="right",width="260"}} 
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[[http://www.math.cas.cz/homepage/main_page.php?id_membre=37Eduard Feireisl]] <<BR>> Professor <<BR>> Institute of Mathematics <<BR>> Czech Academy of Sciences <<BR>> Czech Republic  [[https://www.math.unibielefeld.de/~hofmanova/Martina Hofmanova]] <<BR>> Professor <<BR>> Faculty of Mathematics <<BR>> Bielefeld University <<BR>> Germany 
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Eduard Feireisl is a Professor of Mathematics in the [[http://www.math.cas.czInstitute of Mathematics]] of the [[http://www.cas.czCzech Academy of Sciences]]. He is a renowned specialist in the mathematical theory of fluid dynamics whose expertise has been recognized by many awards, the latest being the Bernard Bolzano Medal.  Martina Hofmanova is a young Professor of Mathematics in the [[https://www.unibielefeld.de/(en)/Bielefeld University]] of Germany. She is a renowned specialist in the mathematical theory of stochastic fluid dynamics, in particular stochastic Navier Stokes and Euler equations. She is also an expert on rough path theory for stochastic partial dierential equations. 
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We consider the phenomenon of oscillations in the solution families to partial differential equations. To begin, we briefly discuss the mechanisms preventing oscillations/concentrations and make a short excursion in the theory of compensated compactness. Pursuing the philosophy "everything what is not forbidden is allowed" we show that certain problems in fluid dynamics admit oscillatory solutions. This fact gives rise to two rather unexpected and in a way contradictory results: (i) many problems describing inviscid fluid motion in several space dimensions admit globalintime (weak solution); (ii) the solutions are not determined uniquely by their initial data. We examine the basic analytical tool behind these rather ground breaking results  the method of convex integration applied to problems in fluid mechanics and, in particular, to the Euler system.  We introduce our model system and main questions of interest. A particular emphasis is put on various notions of solutions. To begin, we briefly discuss the principal concepts from probability theory and stochastic analysis with applications to stochastic PDEs, and make a short excursion in the theory of compressible NavierStokes equations. As the first step in our analysis of the compressible NavierStokes system driven by stochastic forces we establish existence of a dissipative martingale solution. We also show that they satisfy a relative energy inequality and discuss the long time behaviour of dissipative martingale solutions. Next, we prove existence of strong solutions. These solutions are strong in the PDE and probabilistic sense and can only be showed to exist locally in time, that is, up to a positive stopping time. Reference: [[https://www.degruyter.com/view/product/475854Stochastically forced compressible fluid flows, De Gruyter Seriesin Applied and Numerical Mathematics, De Gruyter, 2018.]] 
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* Oscillations, concentrations and how to handle them * Equations preventing oscillations, compactness and compensated compactness * Oscillatory solutions to the compressible Euler system * Oscillatory lemma * Ill posedness of the Euler system in the space dimension N = 2,3 * Extension of the method to other problems in fluid dynamics 
* Stochastic compressible Navier Stokes system. * Weak, Strong, Mild, Stationary solutions to compressible Navier Stokes systems. * Relative energy inequality for compressible fluids with stochastic forcing and their applications. * Markov selection to the stochastic compressible Navier Stokes system. * Ill posedness of the stochastic compressible Euler system. 
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* Januray 22  January 25, 2019 and January 28  February 01, 2019  * February 17  February 21, 2020 and February 24  February 28, 2020 
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* Lecture 1 : 09:30am to 11:00am  * Lecture : 09:30am to 11:00am 
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* Tea Break: 11:00am to 11:30am * Lecture 2 : 11:30am to 01:00pm 

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The audience should be familiar with Sobolev spaces, functional analysis, and general theory of hyperbolic PDEs.  The audience should be familiar with functional analysis, stochastic analysis, and deterministic theory of compressible fluid flows. 
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The course is open to PhD research scholars, postdocs, faculty and scientists working in mathematics departments. Preference will be given to those working on the mathematical aspects of Euler equations.  The course is open to PhD research scholars, postdocs, faculty and scientists working in mathematics departments. Preference will be given to those working on the mathematical aspects of Fluid flows. 
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Deadline for application 21 December, 2018  Announcement of selected participants 24 December, 2018  
Deadline for application 10 January, 2020  Announcement of selected participants 14 January, 2020  
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* Kush Kinra, IIT Roorkee  
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== Outstation Participants ==  * Jisha CR, SRM University 
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== Local Participants ==  * Sudhir Singh, NIT Trichy 
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* Soham Gokhale, IISER Thiruvananthapuram * Divya Jaganathan, ICTSTIFR * Divya Goel, IIT Delhi * Ananta Kumar Majee, IIT Delhi * Neeraj Bhauryal * Saibal Khan * Abhishek * Utsab Sarkar * Adimurthi * Imran Biswas * G D Veerappa Gowda * Mythily Ramaswamy * Ujjwal Koley 
Compact Course on Mathematical aspects of Stochastic Compressible Fluid Flows
Dates : February 17  February 28, 2020 Venue : TIFR Center for Applicable Mathematics, Bangalore
Speaker
Martina Hofmanova
Professor
Faculty of Mathematics
Bielefeld University
Germany
Martina Hofmanova is a young Professor of Mathematics in the Bielefeld University of Germany. She is a renowned specialist in the mathematical theory of stochastic fluid dynamics, in particular stochastic Navier Stokes and Euler equations. She is also an expert on rough path theory for stochastic partial dierential equations.
Objectives
We introduce our model system and main questions of interest. A particular emphasis is put on various notions of solutions. To begin, we briefly discuss the principal concepts from probability theory and stochastic analysis with applications to stochastic PDEs, and make a short excursion in the theory of compressible NavierStokes equations. As the first step in our analysis of the compressible NavierStokes system driven by stochastic forces we establish existence of a dissipative martingale solution. We also show that they satisfy a relative energy inequality and discuss the long time behaviour of dissipative martingale solutions. Next, we prove existence of strong solutions. These solutions are strong in the PDE and probabilistic sense and can only be showed to exist locally in time, that is, up to a positive stopping time.
Course Contents
 Stochastic compressible Navier Stokes system.
 Weak, Strong, Mild, Stationary solutions to compressible Navier Stokes systems.
 Relative energy inequality for compressible fluids with stochastic forcing and their applications.
 Markov selection to the stochastic compressible Navier Stokes system.
 Ill posedness of the stochastic compressible Euler system.
Schedule
 February 17  February 21, 2020 and February 24  February 28, 2020
 Lecture : 09:30am to 11:00am
Prerequisites
The audience should be familiar with functional analysis, stochastic analysis, and deterministic theory of compressible fluid flows.
Who can apply ?
The course is open to PhD research scholars, postdocs, faculty and scientists working in mathematics departments. Preference will be given to those working on the mathematical aspects of Fluid flows.
How to apply ?
Selection will be made based on your CV and research interests. Please send your latest CV and also ask your advisor to send a recommendation letter to conlaw@tifrbng.res.in
Deadline for application 
10 January, 2020 
Announcement of selected participants 
14 January, 2020 
Support
All selected outstation participants will be paid III AC return train fare from their place of study/work to Bangalore. Boarding and lodging in shared accommodations in the hostel will be provided to outstation participants for the period of the workshop.
Organizing Committee
 Ujjwal Koley
 Mythily Ramaswamy
Supported by
 TIFR Centre for Applicable Mathematics, Bangalore
 Airbus Chair on Mathematics of Complex Systems, TIFRCAM, Bangalore
List of Selected Participants
 Kush Kinra, IIT Roorkee
 Jisha CR, SRM University
 Sudhir Singh, NIT Trichy
 Soham Gokhale, IISER Thiruvananthapuram
 Divya Jaganathan, ICTSTIFR
 Divya Goel, IIT Delhi
 Ananta Kumar Majee, IIT Delhi
 Neeraj Bhauryal
 Saibal Khan
 Abhishek
 Utsab Sarkar
 Adimurthi
 Imran Biswas
 G D Veerappa Gowda
 Mythily Ramaswamy
 Ujjwal Koley