AIS Differential Equations (2018)

School Convener(s)

• Scalar conservation laws, hyperbolic equations, weak solutions, Method of characteristics, R-H condition, non-uniqueness of weak solution, entropy conditions, irreversibility

R Radha

• Explained existence and uniqueness of solution to differential equations, the Stability theory for ordinary differential equations, Lyapunov energy functional, one dimensional bifurcation theory, along with types of bifurcations.

• Topics on Hamilton-Jacobi equations, Lagendre transformation, Hopf-Lax formula, Lax-Oleinik formula, Structure theorem for bounded entropy solutions, conservation laws with discontinuous flux, interface entropy condition, uniqueness of (A,B)-entropy solution, L 1 stability

• Lectures on Finite difference equations, consistency, stability of numerical schemes, convergence of numerical solutions, conservative schemes, Lax-Wendroff’s theorem, Lax-Friedrichs scheme, Godunov scheme, Engquist-Osher scheme, monotone schemes, Godunov’s theorem, TVD schemes, l 1contraction schemes, numerical entropy inequality

• Derivation derivation of Euler system for gas dynamics and also given the shallow water wave system.After introducing the method of characteristics, the compatibility condition is derived. Using this,it is shown that the classical solution may not remain constant along the characteristics especially in the case of quasi-linear systems. Taking this as a motivation, Riemann invariants are derived for isentropic flow and shallow water waves. In the case of 2x2 system the invertibility of the map is shown between the solution and the Riemann invariant. Using Riemann invariants, full details of the breakdown of a classical solution in a 2x2 system is discussed and derived sufficient conditions for gradient catastrophe. As an outcome of this derivation, the genuinely nonlinear and linearly degenerate characteristic fields are defined. The theory of simple waves for general systems are discussed where we have characterized the simple waves in terms of the Riemann invariants. Construction of a solution of the Riemann problem for a general system in 1D is explained along with the local existence and uniqueness theorem for a weak solution of the Riemann problem for a general system in one dimensional. Finally, the idea of Glimm approach of constructing approximation solutions to a general initial value problem by using Riemann problem solutions as building blocks is briefly explained.

• lectures on System of conservation laws, R-H condition, Lax-entropy condition, shock and rarefaction waves, entropy solutions of the Cauchy problem, continuous dependence of solutions, uniqueness of entropy solutions for systems and Vanishing viscosity approximation of conservation laws, uniform BV estimates, l ∞ estimates for the solution, uniqueness of an entropy solution respectively.

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