Front profiles in the vanishing-diffusion limit for monostable reaction-diffusion-convection equations

Abstract

We prove the vanishing-viscosity $L^1({\mathbb{R}})$-convergence of minimal-speed travelling-front profiles for scalar balance laws with monostable reaction, possibly non-convex flux, and viscosity $\varepsilon \geq 0$. Such equations are known to admit so-called entropy travelling fronts for all velocities greater than or equal to an $\varepsilon$-dependent minimal value, both for $\varepsilon >0$, when all fronts are smooth, and for $\varepsilon =0$, when the possibly non-convex flux results in fronts of speed close to the minimal value typically having discontinuities where jump conditions hold. Our main result is that, as $\varepsilon \downarrow 0$, profiles of minimal velocity ${c_{\varepsilon}^*}$ converge in $L^1({\mathbb{R}})$ to the unique (up-to-translation) $\varepsilon=0$ entropy-front profile of minimal velocity ${c^*}$. The proofs exploit the compactness inherent in the monotonicity of the profiles, together with uniform-in-$\varepsilon$ estimates on the convergence of the profiles to their spatial limits. Convergence results for the less-delicate fronts of non-minimal speed also follow from the arguments given.