The limit of vanishing viscosity for doubly nonlinear parabolic equations
Electronic Journal of Qualitative Theory of Differential Equations
We show that solutions of the doubly nonlinear parabolic equation ∂b (u) ∂t − div(a(∇u)) + div( f (u)) = g converge in the limit 0 of vanishing viscosity to an entropy solution of the doubly nonlinear hyperbolic equation ∂b (u) ∂t The difficulty here lies in the fact that the functions a and b specifying the diffusion are nonlinear. 2 A. Matas and J. Merker Equation (1.1) may be viewed as a limit of the doubly nonlinear parabolic system ∂b(u) ∂t − div(a(∇u)) + div( f (u)) = g (1.2) for 0, where
... the viscosity tensor a : R m ⊗ R n → R m ⊗ R n converges to zero. Therefore, (1.1) is said to be the limit of vanishing viscosity of (1.2). Note that the viscosity tensor a is allowed to be nonlinear, and even degenerate or singular, thus much more general physical models are covered by (1.2) than by parabolic equations with linear viscosity a(∇u) = ∇u. The aim of this article is to prove convergence of the solutions u of the parabolic equation (1.2) to an entropy solution u of the hyperbolic equation (1.1) as 0. To the best of our knowledge there are no articles which explicitly prove convergence in the case (1.2) of nonlinear diffusion. The scalar case with b(u) = u and nonlinear diffusion is handled in  , however, the proof of the entropy inequality [11, Proposition 3.2] is omitted. The limit of an equation with nonlinear diffusion and dispersion is studied in  . As equation (1.2) is often known to be a good model of a physical system, it is relevant to study the limit of vanishing viscosity for nonlinear diffusions. Therefore, it is interesting to know whether the behaviour of this system is governed by (1.1) when viscosity effects become small or are neglected. There are many articles which prove existence of solutions for (1.1) by different approaches and under more general conditions, e.g. for only continuous f and b with noncontinuous inverse b −1 (see  ), or with general boundary conditions (see [15, 2] ). However, note that it is not obvious how to obtain solutions of (1.1) as a limit of solutions of (1.2) in the case of nonlinear diffusion div(a(∇u)) instead of ordinary viscosity ∆u (as can also be seen from the nontrivial proof of Theorem 1.1). Let us emphasize that for general systems the weak solution of (1.1) selected by the limit of vanishing viscosity may depend on the nonlinear diffusion in the approximating equations. This phenomenon is taken into account by an explicit dependence of the notion of admissible entropies on the functions a and b which specify the diffusion, see Definition 3.1. A similar phenomenon is observed for equations with a discontinuous flux f , where non-equivalent notions of entropy correspond to different applications, see  . However, at least for a scalar equation the admissibility of entropies depends in an obvious way only on b and not on a.