MATHEMATICS

V. F. DYACHENKO

ON THE CAUCHY PROBLEM FOR QUASILINEAR SYSTEMS

(Presented by Academician M. V. Keldysh on 11 VII 1960)

Consider the hyperbolic system of quasilinear equations

\[ \frac{\partial u}{\partial t}+\frac{\partial f(u)}{\partial x}=0, \tag{1} \]

where \(u=\{u_1,u_2,\ldots,u_n\}\), \(f=\{f_1,f_2,\ldots,f_n\}\). Usually a generalized solution of system (1) is understood to be a piecewise-continuous function \(u(x,t)\) satisfying the condition

\[ \oint_{\Gamma} u\,dx-f(u)\,dt=0 \tag{2} \]

for any contour \(\Gamma\). As is known, with this definition the Cauchy problem for system (1) has, generally speaking, a nonunique solution. In the work of I. M. Gel'fand \((^1)\) another definition of a generalized solution is proposed. Namely, instead of system (1) one considers the system with viscosity which approximates it,

\[ \frac{\partial u}{\partial t}+\frac{\partial f(u)}{\partial x} =\varepsilon \frac{\partial}{\partial x} B(u)\frac{\partial u}{\partial x}, \tag{3} \]

where \(\varepsilon\) is a small parameter and \(B(u)\) is an arbitrary positive-definite matrix. The function \(u(x,t)\) is called a generalized solution of system (1) if it is the limit (as \(\varepsilon\to 0\)) of the solutions \(u_\varepsilon(x,t)\) of system (3). We shall show that this definition also does not select a unique solution.

Consider the system of equations

\[ \begin{aligned} \frac{\partial u}{\partial t} -\frac{\partial}{\partial x}\left(\cos\frac{3\pi}{2}\,u\right) &=\varepsilon \frac{\partial}{\partial x}\left(b(u)\frac{\partial u}{\partial x}-\frac{\partial v}{\partial x}\right),\\ \frac{\partial v}{\partial t} -\frac{\partial v^2}{\partial x} &=\varepsilon \frac{\partial}{\partial x}\left(-\frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}\right), \end{aligned} \tag{4} \]

where

\[ b(u)=2-\frac14\,\frac{\cos\left(\frac{3\pi}{2}\,u\right)}{1-u^2}, \]

with initial data

\[ u=\operatorname{th}\frac{4x}{\varepsilon},\qquad v=2\,\operatorname{th}\frac{4x}{\varepsilon}. \tag{5} \]

By direct substitution we verify that the system of functions (5) is a solution of this problem.

On the other hand, consider the system of equations

\[ \begin{aligned} \frac{\partial u}{\partial t} -\frac{\partial}{\partial x}\left(\cos\frac{3\pi}{2}\,u\right) &=\varepsilon \frac{\partial^2 u}{\partial x^2},\\ \frac{\partial v}{\partial t} -\frac{\partial v^2}{\partial x} &=\varepsilon \frac{\partial^2 v}{\partial x^2}. \end{aligned} \tag{6} \]

with the same initial data (5). System (6) consists of two independent equations. Therefore, using the results obtained in papers (¹, ²), one can show that problem (6), (5), for sufficiently small \(\varepsilon\), has a solution close to the following:

\[ u= \begin{cases} \operatorname{sign} x, & \text{for } \left|\dfrac{x}{t}\right|>a,\\[6pt] \dfrac{2}{3\pi}\arcsin\left(\dfrac{2}{3\pi}\dfrac{x}{t}\right), & \text{for } \left|\dfrac{x}{t}\right|<a; \end{cases} \qquad v=2\operatorname{sign}x, \tag{7} \]

where \(a\) is the smallest positive root of the equation

\[ a^2\left(\left(\frac{2}{3\pi}\right)^2+ \left(1-\frac{2}{3\pi}\arcsin\frac{2a}{3\pi}\right)^2\right)=1. \]

Systems (4) and (6) are evolutionary; they approximate the same system of hyperbolic equations

\[ \frac{\partial u}{\partial t} - \frac{\partial}{\partial x}\left(\cos\frac{3\pi}{2}u\right)=0, \]

\[ \frac{\partial v}{\partial t} - \frac{\partial v^2}{\partial x}=0 \]

and differ only in the viscosity matrices \(B(u)\). Nevertheless, their limiting solutions (as \(\varepsilon\to0\)) clearly do not coincide. One is

\[ u=\operatorname{sign}x,\qquad v=2\operatorname{sign}x, \]

and the other is (7).

In constructing the viscosity matrix in system (4), ideas contained in the work of S. K. Godunov (³) were used.

I take this opportunity to express my gratitude to I. M. Gelfand for valuable discussions.

CITED LITERATURE

¹ I. M. Gelfand, Uspekhi Mat. Nauk, 14, no. 2 (86) (1959).
² A. S. Kalashnikov, DAN, 127, no. 1 (1959).
³ S. K. Godunov, DAN, 134, no. 6 (1960).