Scalar conservation laws in one space dimension with strict convex flux

scalar conservation

The Department of Mathematics is arranging the fifth instalment of the Departmental Distinguished Lecture Series on April 25, 2022, at 3.45 pm IST. Reputed Mathematician, Prof A Adimurthi will deliver an informative lecture on the topic ‘Controllability of Scalar Conservation Laws with Discontinuous Flux’.

About the Talk

Navier-Stokes and Euler equations play an important role in studying the flow of incompressible fluids. Weak solutions to these equations can be obtained by Galerkin method but the uniqueness is a big open problem. It is a big challenge to obtain an extra condition for the class of functions, so that in this class obtain the existence and uniqueness. To understand this phenomenon, it is better to look at a one-dimensional case where the equation turns out to be viscous Burger’s equation or Burger’s equation with non-linearity is of quadratic order. In this talk, we will restrict to Burger’s type equations called the Scalar Conservation Laws in one space dimension with strict convex flux. Way back in the 50’s, this equation was studied by Lax and Oleinik and obtained an explicit formula for a solution. Oleinik showed that this satisfies an extraction called the Entropy Condition and then showed that in this class, the solution is unique. Later Kruzkov, in an ingenious way, generalized this to obtain a unique solution for Scalar Conservation Law in higher dimensions and Lipschitz fluxes. This result was taken up by Zuazua and his collaborators and studied the Optimal Controllability for Burgers equation. They showed the existence of optimal control and to capture this, they derived a numerical algorithm whose convergence is still open. In a different direction, this was attacked, and the problem was completely solved. Getting the optimal solution is via projection method in a Hilbert space. Recently, this was extended in a non-trivial way to conservation laws with convex discontinuous flux. The talk will explain the main ideas of this work.

About the Speaker

A mathematician of repute, Prof A Adimurthi has been associated with the Tata Institute of Fundamental Research, Centre for Applicable Mathematics Bangalore. At present, he is working as a visiting professor at the Department of Mathematics and Statistics, IIT Kanpur. He has authored more than a hundred research articles on a wide range of topics including Partial differential equations, Functional analysis, Numerical analysis, Calculus of variations, Optimal control, and many others. Prof Adi Adimurthi has been elected as a fellow of; The Indian Academy of Sciences (F.A.Sc) Bengaluru (1996), The National Academy of Sciences (F.N.A.Sc), India in 1997, The Indian National Science Academy (F.N.A) in 1998, The World Academy of Sciences(TWAS) in 2018, Raja Ramanna and DST J.C. Bose National. Along with these fellowships, Professor Adimurthi has also been honoured with the prestigious Dr Zakir Hussain Award in 2013.

Join this informative talk with Prof A Adimurthi on April 25, 2022 at 3.45 pm IST.

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