2. Viscous CONSERVATION LAWS Consider the Cauchy problem of a general system of viscous conservation laws: (2.1) ut + f(u)x = {B{u)ux)x, -o o x oo, t 0, (2.2) u(x,0) = UQ(X), oo x oo, where u = it(x, t) is an n-vector, f(u) is a smooth n-vector valued function, B(u) a smooth n x n matrix, and UQ(X) a given n-vector valued function. We assume that UQ is a small perturbation of a constant state. Without loss of generality, we assume that the constant state is zero. To assure that system (2.1) is dissipative we make the following three basic assumptions: Assumption 2.1. System (2.1) has a strictly convex entropy U ([Sm], [K]). Assumption 2.1 means that there exist a pair of smooth functions U(u) and F(u) (called entropy pair), such that U is strictly convex, (V£/)/' = V F , and (\/2U)B is symmetric and semi-positive definite, where / ' is the Jacobi matrix of / , and V2U is the Hessian of U. We set AQ(U) = V2U(u). Clearly AQ is symmetric and positive definite. Assumption 2.2. For small u, B(u) ^ 0. There exists a smooth one-to-one mapping u fo(y), /o(0) = 0, such that the null space J\f of B(u) = B(fo(u))fb(u) is independent of u. Moreover, Ar_L is invariant under /Q( , U) £ AO(/O(^)), and B{u) maps E n into A/""1, where M1- is the orthogonal completement of J\f. Assumption 2.3. Any eigenvector of /'(0) is not in the null space of B(0). We notice that several interesting physical models, e.g. the Navier-Stokes equations and the equations of magnetohydrodynamics, satisfy Assumptions 2.1-2.3 (c.f. Section 9). We also notice that if B(u) is nonsingular, Assumptions 2.2 and 2.3 are automatically satisfied. Under Assumption 2.1, it follows that A$f is symmetric [FL]. Since Ao is symmet- i ric and positive definite, there exists a symmetric positive definite matrix AQ such that i L i . AQ = ( A Q ) 2 . Thus A Q / ^ A Q ) - 1 is symmetric. Consequently, all the eigenvalues of AQ / ' ( A p ) - 1 are real, and AQ f '(AQ)~1 has a complete set of eigenvectors. Notice that if / ' ( A Q ) - 1 has an eigenvalue A and a corresponding eigenvector r, A 0 2 / / (AQ) _ 1 r = Ar, 7
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