SBV Regularity of Entropy Solutions for a Class of Genuinely Nonlinear Scalar Balance Laws With Non-Convex Flux Function

Abstract

In this work we study the regularity of entropy solutions of the genuinely nonlinear scalar balance laws We assume that the source term g C1( × × +), that the flux function f C2( × × +) and that ui : fuu(ui,x,t) = 0 is at most countable for every fixed (x,t) Ω. Our main result, which is a unification of two proposed intermediate theorems, states that BV entropy solutions of such equations belong to SBVloc(Ω). Moreover, using the theory of generalized characteristics we prove that for entropy solutions of balance laws with convex flux function, there exists a constant C > 0 such that: where C can be chosen uniformly for (x +h,t), (x,t) in any compact subset of Ω.

SBV Regularity for Hamilton–Jacobi Equations in $${{\mathbb R}^n}$$

“…Theorem 1.1 was proved first by Luigi Ambrosio and the second author in the special case n = 1 (see [3] and also [13] for the extension to Hamiltonians H depending on (t, x) and u). Some of the ideas of our proof originate indeed in the work [3].…”

On the attainable set for scalar balance laws with distributed control

“…Regarding a general existence theorem for (1.1)−(1.2), we refer to [19] again, while systems of balance laws, e.g., are treated in [1]. Moreover, to our knowledge, in literature smoothness assumptions on z are usually required in order to get a priori estimates on the positive waves [23]. Indeed, such a priori estimate is the main ingredient of the following…”

Lower compactness estimates for scalar balance laws

“…An inequality of this kind was established in [13,Theorem 1.2]. Here, we provide a slightly more accurate estimate, determining how the constant C appearing in [13, Theorem 1.2] depends on the time t and on the set of points x, y for which the inequality holds.…”