Viscous Profiles of Traveling Waves in Scalar Balance Laws: The Canard Case

Abstract

The traveling wave problem for a viscous conservation law with a nonlinear source term leads to a singularly perturbed problem which necessarily involves a non-hyperbolic point. The correponding slow-fast system indicates the existence of canard solutions which follow both stable and unstable parts of the slow manifold.In the present paper we show that for the viscous equation there exist such heteroclinic waves of canard type. Moreover, we determine their wave speed up to first order in the small viscosity parameter by a Melnikov-like calculation after a blow-up near the non-hyperbolic point. It is also shown that there are discontinuous waves of the inviscid equation which do not have a counterpart in the viscous case.

The stability of travelling fronts for general scalar viscous balance law

“…Considering the cases with sufficiently small viscosity and comparing our results with the results for inviscid case in [2,6,8], one can easily understand the following fact: There is a critical speed separating the continuous profiles from the discontinuous ones of (1.2), as was shown in [8], where the critical speed c cri = min [u k ,u j ] f (u). When ε = 0, it is easy to check that in nonconvex convection cases (f (u k ) may not equal to min [u k ,u j ] f (u)), only for speeds c < c cri there exist smooth monotone waves (iii) of (1.2), no smooth waves exist for the case c cri c c * 0 with c * 0 satisfying min [u k ,u j ] f (u) c * 0 f (u k ).…”

“…Combining the results in [2], we can immediately obtain Corollary 2.1. Let (I) and (II) hold, for any wave U 0,c (x − ct) connecting u k and u j of (1.2), where u k , u j , and c are in one of the following cases:…”

“…In fact, the methods and proof in [2] are still valid for the nonconvex convection cases, although the author did not mention that.…”

“…), while for hyperbolic conservation laws the only travelling waves are shock waves. More recently, when the convection term f is convex, by invariant manifold theory J. Härterich [2] showed that several types of waves of (1.2) obtained in [6] admit the viscous profiles of travelling waves of (1.1), and they are close in weighted space L 1 β (R) when the viscosity ε is small enough. In particular, the author proved that for every continuous wave u 0,c (x − ct) connecting u k to u j of (1.2) in the following four cases:…”

“…where the reaction term g(u) describing some chemical reactions, combustion or other interactions [2] can dramatically change the long time behavior of solutions of (1.2), compared to hyperbolic conservation laws. For example, C. Mascia [6] showed that for the case f is convex there exist many types of travelling waves for (1.2), including the continuous monotone waves and many kinds of discontinuous waves (shock waves, sub-shocks, etc.…”