Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation

Yoshikawa, Atsushi

doi:10.1090/S0002-9947-1993-1208881-X

Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation
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Trans. Amer. Math. Soc. 340 (1993), 103-126 Request permission

Abstract:

It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data ${\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf {R}}^n}, \eta \ne 0$. Here ${r_1}(\eta )$ is a characteristic vector, and ${a_0}(\sigma ,x)$ is a smooth scalar function of compact support. Under the additional requirements that $n = 2$ or $3$ and that ${a_0}(\sigma ,x)$ have the vanishing mean with respect to $\sigma$, it is shown that a genuine solution exists in a time interval independent of $\lambda$, and that the formal solution is asymptotic to the genuine solution as $\lambda \to \infty$.

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