Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


Authors: Jie Jiang and Xue Kong Wang
Journal: Quart. Appl. Math. 50 (1992), 547-557
MSC: Primary 35L65; Secondary 34B15, 34E15
DOI: https://doi.org/10.1090/qam/1178434
MathSciNet review: MR1178434
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Abstract: This paper deals with the singular perturbation boundary value problem

$\displaystyle \left\{ {_{v\left( { - \infty } \right) = A, \qquad v\left( { + \... ...eft( s \right) + f\left( {v\left( s \right)} \right) = 0 \qquad in R,}} \right.$

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

$\displaystyle \left\{ {\begin{array}{*{20}{c}} { - {{\left( {\frac{{w'\left( t ... ...}\\ {w\left( A \right) = 0, \qquad w\left( B \right) = 0.} \end{array}} \right.$

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

$\displaystyle {v_0}\left( s \right) = A + \left( {B - A} \right)H\left( {s - \frac{{\Phi \left( B \right)}}{{G\left( B \right)}}} \right), \qquad s \in R$

, 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 \mathord{\left/ {\vphantom {x t}} \right. \kern-\nulldelimiterspace} t}} \right)$, as a solution to the Riemann problem

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial G\left( u \righ... ...eft( {B - A} \right)H\left( x \right), \qquad for x \in R} \end{array}} \right.$

pointwise converges to $ {u_0}\left( {x, t} \right) = :{v_0}\left( {{x \mathord{\left/ {\vphantom {x t}} \right. \kern-\nulldelimiterspace} 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)} \mathord{\left/ {\vphantom {{\left( B \right)} {g\left( B \right)}}} \right. \kern-\nulldelimiterspace} {g\left( B \right)}}$, and the restriction on $ \Phi \left( u \right)$ and $ G\left( u \right)$,

$\displaystyle \Phi \left( u \right) - G\left( u \right)\Phi {{\left( B \right)}... ...n-\nulldelimiterspace} G}\left( B \right) \ge 0 \qquad on\left[ {A, B} \right],$

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.

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DOI: https://doi.org/10.1090/qam/1178434
Article copyright: © Copyright 1992 American Mathematical Society


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