The L² theory of well-posedness for hyperbolic systems of conservation laws

In this dissertation, we describe new developments in the L² theory for the well-posedness of hyperbolic systems of conservation laws. We use the theory of shifts (see Vasseur [Handbook of Differential Equations: Evolutionary Equations, 4:323-376, 2008]) and a-contraction (see Kang and Vasseur [Arch. Ration. Mech. Anal., 222(1):343-391, 2016]). The advantage to the L² theory is that it is not perturbative. There are no small data limitations. The results in this dissertation are able to shed new light on the uniqueness and stability of solutions to hyperbolic systems of conservation laws with very large data.