A singular perturbation nonlinear boundary value problem and the $E$-condition for a scalar conservation law

This paper deals with the singular perturbation boundary value problem \[ \left \{ {_{v\left ( { - \infty } \right ) = A, \qquad v\left ( { + \infty } \right ) = B; \qquad \varepsilon \ge 0, \qquad A < B}^{\varepsilon {{\left ( {k\left ( {v\left ( s \right )} \right )v’\left ( s \right )} \right )}’} + \left ( {sg\left ( {v\left ( s \right )} \right ) - \varphi \left ( {v\left ( s \right )} \right )} \right )v’\left ( s \right ) + f\left ( {v\left ( s \right )} \right ) = 0 \qquad \mathrm{in}\; R,}} \right .\] whose solution ${v_\varepsilon }\left ( s \right )$ is constructed by the aid of the solution ${w_\varepsilon }\left ( t \right )$ to the two-point boundary value problem \[ \left \{ {\begin {array}{*{20}{c}} { - {{\left ( {\frac {{w’\left ( t \right ) - \varphi \left ( t \right ) + \varepsilon f\left ( t \right )k{{\left ( t \right )} \left / {\vphantom {{\left ( t \right )} {w\left ( t \right )}}} \right . {w\left ( t \right )}}}}{{g\left ( t \right )}}} \right )}’} = \frac {{\varepsilon k\left ( t \right )}}{{w\left ( t \right )}} \;\mathrm{in}\; \left ( {A, B} \right ),}\\ {w\left ( A \right ) = 0, \qquad w\left ( B \right ) = 0.} \end {array}} \right .\] The restrictions on $\varphi \left ( t \right )$, $g\left ( t \right )$, $k\left ( t \right )$, and $f\left ( t \right )$ not only ensure that the two-point boundary value problem has a solution ${w_\varepsilon }\left ( t \right )$ but also guarantee that as $\varepsilon$ tends to zero the solution ${w_\varepsilon }\left ( s \right )$ pointwise converges to \[ {v_0}\left ( s \right ) = A + \left ( {B - A} \right )H\left ( {s - \frac {{\Phi \left ( B \right )}}{{G\left ( B \right )}}} \right ), \qquad s \in R\], the solution to the reduced problem, where $H\left ( s \right )$ is the multiple-valued Heaviside function, $G\left ( t \right ) =:\int _A^{t} g\left ( s \right ) ds$, and $\Phi \left ( t \right ) =:\int _{A}^{t} \varphi \left ( s \right ) ds$. Moreover, the function ${u_\varepsilon }\left ( {x, t} \right ) = :{v_\varepsilon }\left ( {{x \left / {\vphantom {x t}} \right . t}} \right )$, as a solution to the Riemann problem \[ \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial G\left ( u \right )}}{{\partial t}} + \frac {{\partial \Phi \left ( u \right )}}{{\partial x}} = \frac {{f\left ( u \right )}}{t} + \varepsilon t\frac {\partial }{{\partial x}}\left ( {k\left ( u \right )\frac {{\partial u}}{{\partial x}}} \right ), \qquad x \in R, \qquad t > 0,}\\ {u\left ( {x, 0} \right ) = A + \left ( {B - A} \right )H\left ( x \right ), \qquad for x \in R} \end {array}} \right .\] pointwise converges to ${u_0}\left ( {x, t} \right ) = :{v_0}\left ( {{x \left / {\vphantom {x t}} \right . t}} \right )$, the discontinuous solution of the Riemann problem for the scalar conservation law $\left ( {\varepsilon = 0} \right )$. Obviously, ${u_0}\left ( {x, t} \right )$ satisfies the classical Rankine-Hugoniot condition on the line of discontinuity $x = \\ t\Phi {{\left ( B \right )} \left / {\vphantom {{\left ( B \right )} {g\left ( B \right )}}} \right . {g\left ( B \right )}}$, and the restriction on $\Phi \left ( u \right )$ and $G\left ( u \right )$, \[ \Phi \left ( u \right ) - G\left ( u \right )\Phi {{\left ( B \right )} \left / {\vphantom {{\left ( B \right )} G}} \right . G}\left ( B \right ) \ge 0 \qquad on\left [ {A, B} \right ],\] is exactly the E-Condition proposed first by Oleinik. The technical arguments, which involve only the use of the Schauder Fixed Point Theorem and integral representations, are elementary.