Abstract:
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. In order 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 Olenik and obtained an explicit formula for a solution. Olenik 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 who 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. In this talk, I will explain the main ideas of this work.