The Riemann problem for general 2×2 conservation laws
The Riemann problem for general 2×2 conservation laws
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一般 2×2 守恒定律的黎曼问题
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发表时间：
1974
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通讯作者：
Tai
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作者：
Tai
具任意初值状态(2)(u0(*))的双曲型守恒律组ut+f(u，v)=0，(1)‘vt+g(u，v)x=0的Riemann问题。O<；*»í(«/.V‘)，\(.ur，VR)，x<；0，x>；0。我们假设Fv<；0，Gu<；0。设L(R)是df(f，g)的左(右)特征向量，d(f，g)是(u，v)平面上一维流形的不相交并，而不是假定通常的凸性条件d(R)i*0，1？1，2.将单方程的Oleinik条件(E)推广到系统(1)，又称这个新条件(E)。我们的条件(E)包含Lax的冲击不等式，并且当d(R)元=0时，两者等价。然后证明了黎曼问题(1)和(2)在满足条件(E)的激波、稀疏波和接触不连续类中存在唯一解。导言。我们考虑系统ut+fiu，v)r=0，(o.i)*jy)xt>；o，-~<；x<；oo，vt+g(u，v)x=0，其中u=m(x，f)，v=u(jc，f)，且对于P2中的某个开集U，g E C3(F7).要解决的问题是对任意常数Iuv)E U，(R，Vr)E U的Riemann问题{(；，v‘)；Iur，Vr)}，即用初始数据(0.2)(Uix，0)，v(x，0)=iu0ix)，对于x>；0，求解系统(0.1))={×Gb^。1972年1月26日提交给学会；1973年10月16日由编辑收到。AMS(MOS)主题分类(1970)。主35L65、35F25。
The Riemann Problem for a system of hyperbolic conservation laws of form ut + f(u, v) = 0, (1) ' vt + g(u, v)x = 0 with arbitrary initial constant states (2) (u0(*). «o<*» í(«/. v¡), \(.ur, vr), x<0, x > 0, is considered. We assume that fv < 0, gu < 0. Let l¡ (r¡) be the left (right) eigenvectors of dF ■ d(f, g) for eigenvalues \j < \2. Instead of assuming the usual convexity condition d\¡(r¡) i* 0,1 « 1, 2, we assume that d\¡(r¡) = 0 on disjoint union of 1-dim manifolds in the (u, v) plane. Oleinik's condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax's shock inequalities and, in case d\¡(r¡) ¥= 0, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E). Introduction. We consider the system ut + fiu, v)r = 0, (o.i) *Jy )x t>o,-~<x<oo, vt+g(u,v)x = 0, where u = m(x, f), v = u(jc, f) and /, g E C3(f7) for some open set U in P2. The problem to be solved is the Riemann Problem {(«;, v¡); iur, vr)} for arbitrary constants iu¡v¡) E U, («r, vr) E U; i.e. solve the system (0.1) with initial data (0.2) (uix, 0), v(x, 0)) = iu0ix), v0ix)) = {¡£ ^ for jc<0, for x > 0. Presented to the Society, January 26, 1972; received by the editors October 16, 1973. AMS (MOS) subject classifications (1970). Primary 35L65, 35F25.