Weak solutions for contractive nonlinear equations and parabolic relaxation limits

nonlinear contractive equations: “dissipative solution”. We show that this notion is equivalent to two previously defined. One is the viscosity solution for Hamilton-Jacobi equations and fully nonlinear elliptic equations, as introduced by Crandall-Lions. The other is the entropy solution for conservation laws introduced by Kružkov. The proposed notion is based on properties of accretive operators.; For conservation laws, dissipative solutions are much easier to work with than Kružkov's (equivalent) entropy solutions. To illustrate this, we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.