Convex integration for scalar conservation laws in one space dimension

In the absence of the entropy condition, we construct an L^∞ solution to the Cauchy problem of a scalar conservation law in one space dimension that exhibits fine-scale oscillations in the interior of its support when the initial function is nonconstant, continuous, and compactly supported. As a result, such a solution turns out to be nowhere continuous in the interior of the support. The method of proof is to convert the main equation into a suitable partial differential inclusion and to rely on the convex integration method of Müller and Šverák [Ann. of Math. (2), 157 (2003), pp. 715–742]. In doing so, we find an appropriate subsolution by solving certain ordinary differential equations and make use of it to tailor an in-approximation scheme that reflects persistence of oscillations.