Optimal Decay Exponents for Fractional Burgers Rarefaction Waves

Abstract

We establish decay estimates toward rarefaction waves for the one-dimensional fractional Burgers equation when the order of dissipation lies between one and two. The initial data are bounded, and their perturbation from the Riemann step belongs to the critical weak-Lorentz space determined by scaling; the perturbation need not be small or integrable. We obtain optimal algebraic decay exponents, improving the known decay rates, and extending convergence to every Lebesgue space above the critical exponent, including the supremum norm. We prove optimality by constructing a single solution with Riemann step initial data whose lower decay estimates have exactly the same algebraic exponents as our upper estimates. These lower bounds establish that neither the algebraic decay exponents nor the range of convergence can be improved under the stated assumptions. The upper estimates contain a logarithmic factor whose necessity remains open. The proof combines convex truncation estimates, fractional interpolation, and a one-sided slope estimate for the supremum norm.