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Asymptotic expansions of the solutions to a class of quasilinear hyperbolic initial value problems. (English) Zbl 0855.35076

The author studies the initial value problem related to the following quasilinear positive strictly hyperbolic system \[ A_0(u) \partial_t u+ \sum^n_{\nu= 1} A_\nu(u) \partial_{x_\nu} u+ B(u) u= 0.\tag{1} \] Here \(A_0(u), A_1(u),\dots, A_n(u)\) are \((m\times m)\) symmetric matrices depending smoothly on \(u\in \mathbb{R}^m\), and \(A_0(u)\) is positive definite. The initial data are of the form \[ u= \lambda^{- 1} g(\lambda x \eta, x),\quad t= 0,\tag{2} \] where \(\lambda> 0\) is a large parameter and the function \(g\) satisfies \(\int_{\mathbb{R}} g(s, x) ds= 0\). Strict hyperbolicity guarantees that the matrix \(M(u, \xi)= \sum^n_{\nu= 1} \xi_\nu A_0(u)^{- 1} A_\nu(u)\), \(\xi= (\xi_1,\dots, \xi_n)\neq 0\), has \(m\) distinct real eigenvalues \(p_1(u, \xi),\dots, p_m(u, \xi)\). Denote by \(r_1(u, \xi),\dots, r_m(u, \xi)\) the corresponding eigenvectors, and let \(X_j= {^t r}_j(u, \eta) \nabla_u\), \(i= 1,\dots, m\), be the corresponding characteristic vector fields. The main assumption concerning these fields is that the commutator of \(X_j\) and \(X_k\) is a linear combination of the same fields \(X_j\) and \(X_k\).
The first main result asserts that for the case of space dimensions \(n= 2,3\) the corresponding Cauchy problem (1), (2) has a local solution in the space \[ u(\cdot, t, \lambda)\in L^\infty([0, T_1];\;H^3(\mathbb{R}^n))\cap C([0, T_1];\;H^\sigma(\mathbb{R}^n)), \]
\[ \partial_t u(\cdot, t, \lambda)\in L^\infty([0, T_1];\;H^2(\mathbb{R}^n))\cap C([0, T_1]; H^{\sigma- 1}(\mathbb{R}^n)) \] for any \(\sigma< 3\). Here the number \(T_1\), defining the interval of existence of the solution, is independent of the parameter \(\lambda\).
The next step is the construction of an asymptotic solution of the form \[ U = \lambda^{- 1} \sum^m_{j= 1} a_j(\lambda S_j(x, t), x, t) r_j(0, \eta)+\lambda^{- 2} \sum^m_{i, j, k= 1} b_{ijk} (\lambda S_i(x, t), \lambda S_j(x, t), x, t) r_k(0, \eta). \] Here the phase functions \(S_j(x, t)\) are determined by \[ S_j(x, t)= - p_j(0, \eta) t+ x \eta,\quad j= 1,\dots, m, \] while \(a_j(s_i, x, t)\), \(b_{ijk}(s_i, s_j, x, t)\) satisfy suitable transport equations. An estimate of type \[ |u(\cdot, t, \lambda)- U(\cdot, t, \lambda)|_s\leq K\lambda^{s- 3},\quad 0\leq s\leq 3, \] where \(|\;|_s\) is the Sobolev norm, is established.
Reviewer: V.Georgiev (Sofia)

MSC:

35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35C20 Asymptotic expansions of solutions to PDEs

Keywords:

local solution
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