标题： TVD场和等熵气体流动 显示英文标题

标题： TVD Fields and Isentropic Gas Flow

摘要： 关于一维空间守恒律方程组柯西问题的大变差解的整体存在性知之甚少。除了通过补偿紧性方法得到的 $L^\infty$ 数据的结果外，对于任意BV数据全局BV解的存在性仍然是一个突出的开放问题。 特别是，目前尚不清楚等熵气体动力学是否满足适用于所有BV数据的先验变差界。 在少数情况下，此类结果是可用的：纯量方程，Temple类系统，满足Bakhvalov条件的 $2\times 2$-系统，以及特别是等温气体动力学。 在这每一种情况下，方程都允许一个TVD（总变差递减）场：定义在状态空间上的一个标量函数，其沿熵解的空间变差不会随时间增加。 在本文中，我们考虑严格的双曲 $2\times 2$-系统，并在Glimm方案下解决这些波相互作用时，推导出跨越所有成对波相互作用的标量场的表示结果。 然后我们利用这一点来证明具有 $\gamma$-定律压力函数的等熵气体动力学不承认任何非平凡的这种类型的TVD场。 显示英文摘要

摘要： Little is known about global existence of large-variation solutions to Cauchy problems for systems of conservation laws in one space dimension. Besides results for $L^\infty$ data via compensated compactness, the existence of global BV solutions for arbitrary BV data remains an outstanding open problem. In particular, it is not known if isentropic gas dynamics admits an a priori variation bound which applies to all BV data. In a few cases such results are available: scalar equations, Temple class systems, $2\times 2$-systems satisfying Bakhvalov's condition, and, in particular, isothermal gas dynamics. In each of these cases the equations admit a TVD (Total Variation Diminishing) field: a scalar function defined on state space whose spatial variation along entropic solutions does not increase in time. In this paper we consider strictly hyperbolic $2\times 2$-systems and derive a representation result for scalar fields that are TVD across all pairwise wave interactions, when the latter are resolved as in the Glimm scheme. We then use this to show that isentropic gas dynamics with a $\gamma$-law pressure function does not admit any nontrivial TVD field of this type.