Quasi-Linear Parabolic Systems

We have studied second-order quasi-linear parabolic systems in Chapter 14, where it was assumed that the equations admitted a bounded invariant region. For the gas dynamics equations with all of the dissipative mechanisms taken into account (viscosity and thermal conductivity), and for various models of these, there may exist invariant regions, but they are usually unbounded. Thus we cannot conclude that the solution is a-priori bounded, and global existence theorems become more difficult to prove. One way to overcome this problem is to obtain “energy” inequalities in the unknown function and its derivatives, in a manner somewhat analogous to what we have done for linear hyperbolic equations in Chapter 4. In order to obtain these estimates for nonlinear equations, certain additional restrictions must be imposed : small data, special forms of the equations, restrictions on the data at infinity, and so on.