2024/11/20 by Debora Amadori, Alberto Bressan, Amadori, Debora +3 · 1 citation
Engineering · Mathematics · #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #Gas Dynamics and Kinetic Theory
paper · pdf · doi:10.4310/cms.260715020746
The paper is concerned with a scalar conservation law with discontinuous gradientdependent flux. Namely, the flux is described by two different functions f(u) or g(u), when the gradient ux of the solution is positive or negative, respectively. We study here the unstable case where f(u)>g(u) for all u ∈ ℝ. Assuming that both f and g are strictly convex, solutions to the Riemann problem are constructed. Even for a smooth initial data, examples show that the Cauchy problem can have infinitely many solutions. For an initial data which is piecewise monotone, i.e., increasing or decreasing on a finite number of intervals, a solution can be constructed globally in time. It is proved that such solution is unique under the additional requirement that the number of interfaces, where the flux switches between f and g, remains as small as possible.