Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic behavior of traveling wave solutions of the equations for the flow of a fluid with small viscosity and capillarity

Author: J. L. Boldrini
Journal: Quart. Appl. Math. 44 (1987), 697-708
MSC: Primary 35B40; Secondary 35Q20, 76D99
DOI: https://doi.org/10.1090/qam/872822
MathSciNet review: 872822
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Abstract: We study the oscillations of the traveling wave solutions of

$\displaystyle \left\{ {_{{v_t} = - p{{\left( u \right)}_x} + \epsilon {v_{xx}} - \delta {u_{xxx}}}^{{u_t} = {v_x},}} \right.$

for small $ \epsilon $ and $ \delta $. These solutions give information about the structure of the shock layers in fluids with small viscosity and capillarity. We conclude that the traveling wave has oscillations with increasing amplitude when $ \epsilon $ and $ \delta $ approach zero such that $ \delta \ne O\left( {{\epsilon ^2}} \right)$. When $ \delta = o\left( {{\epsilon ^2}} \right)$, if there are oscillations, their amplitude decreases to zero as $ \epsilon $ and $ \delta $ approach zero. When $ \delta = {\epsilon ^2}$ the shape of the traveling wave is independent of the magnitude of $ \epsilon $ and $ \delta $.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/qam/872822
Article copyright: © Copyright 1987 American Mathematical Society

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