Duration: 16 hours
Linear Theory: Spaces involving time; Second-order hyperbolic equations, hyperbolic systems of first-order equations, examples; Weak solutions, well-posedness; Galerkin method, Vanishing viscosity method, , energy methods, Fourier transform method.
Nonlinear Theory II – One-Dimensional Systems of Nonservation Laws: Riemann problem, Cauchy problem; Elementary waves: shock waves, rarefaction waves, contact discontinuities; Lax entropy conditions; Glimm scheme, front-tracking, BV solutions; *Compensated compactness, entropy analysis, Lᵖ solutions, vanishing viscosity methods; *Uniqueness and continuous dependence.
Nonlinear Theory III – Noninear Wave Equations: Local existence and energy estimates, Galerkin method; Global existence of semi-linear wave equations with small data (Quasilinear case could be similarly treated); Lower regularity results for large data; *Littlewood-Paley theory and Strichartz estimates.
*Nonlinear Theory IV - Multidimensional Systems of Conservation Laws: Basic features/phenomena (re-visit); Local existence and stability; formation of singularities; Discontinuities and free boundary problems; Stability of shock waves, rarefaction waves, vortex sheets, entropy waves.
Prerequisites: Introduction to PDE foundation module and Analsyis of PDEs, parts 1 and 2
Homework Problem Sets: