ON THE EXISTENCE AND COMPACTNESS OF A TWO DIMENSIONAL RESONANT SYSTEM OF CONSERVATION LAWS

ON THE EXISTENCE AND COMPACTNESS OF A TWO-DIMENSIONAL RESONANT SYSTEM OF CONSERVATION LAWS KENNETH H. KARLSEN, MICHEL RASCLE, AND EITAN TADMOR Abstract. We prove the existence of a weak solution to a two-dimensional resonant 3? 3 system of conservation laws with BV initial data. Due to possible resonance (coinciding eigenvalues), spatial BV estimates are in general not available. Instead, we use an entropy dissipation bound combined with the time translation invariance property of the system to prove existence based on a two-dimensional compensated compactness argument adapted from [36]. Existence is proved under the assumption that the flux functions in the two directions are linearly independent. 1. Introduction This paper studies certain two-dimensional resonant 3? 3 systems of conservation laws of the form kt = 0, lt = 0, ut + f(k, u)x + g(l, u)y = 0, (1.1) which are augmented with L∞ ?BV initial data k|t=0 = k(x, y), l|t=0 = l(x, y), u|t=0 = u0(x, y).(1.2) The goal is to prove that there exists a weak solution to (1.1)–(1.2). In recent years the one-dimensional version of the above system, kt = 0, ut + f(k, u)x = 0, (1.3) has received a considerable amount of attention.

• bv initial


• notation let


• let ? ?


• mapping ap- proach


• continuous mapping


• spatial bv compactness


• lemma


• conservation laws


• dimensional