Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 

 

Global asymptotics toward the rarefaction wave for solutions of viscous $ p$-system with boundary effect


Authors: Akitaka Matsumura and Kenji Nishihara
Journal: Quart. Appl. Math. 58 (2000), 69-83
MSC: Primary 35B40; Secondary 35L50, 76L05, 76N10
DOI: https://doi.org/10.1090/qam/1738558
MathSciNet review: MR1738558
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The initial-boundary value problem on the half-line $ {R_ + } \cdot = \left( 0, \infty \right)$ for a system of barotropic viscous flow $ {v_t} - {u_x} = 0, {u_t} + p{\left( v \right)_x} = \mu {\left( {\frac{{{u_x}}}{v}} \right)_x}$ is investigated, where the pressure $ p\left( v \right) = {v^{ - \gamma }}\left( \gamma \ge 1 \right)$ for the specific volume $ v > 0$. Note that the boundary value at $ x = 0$ is given only for the velocity $ u$, say $ {u_ - }$ and that the initial data $ \left( {v_0}, {u_0} \right)\left( x \right)$ have the constant states $ \left( {v_ + }, {u_ + } \right)$ at $ x = + \infty $ with $ {v_0}\left( x \right) > 0, {v_ + } > 0$. If $ {u_ - } < {u_ + }$, then there is a unique $ {v_ - }$ such that $ \left( {v_ + }, {u_ + } \right) \in {R_2}\left( {v_ - }, {u_ - } \right)$ (the 2-rarefaction curve) and hence there exists the 2-rarefaction wave $ \left( v_2^R, u_2^R \right)\left( x/t \right)$ connecting $ \left( {v_ - }, {u_ - } \right)$ with $ \left( {v_ + }, {u_ + } \right)$. Our assertion is that, if $ {u_ - } < {u_ + }$, then there exists a global solution $ \left( v, u \right)\left( t, x \right)$ in $ {C^0}\left( \left[ 0, \infty \right);{H^1}\left( {R_ + } \right) \right) $, which tends to the 2-rarefaction wave $ \left( v_2^R, u_2^R \right)\left( x/t \right)\left\vert {_{x \ge 0}} \right.$ as $ t \to \infty $ in the maximum norm, with no smallness condition on $ \left\vert {{u_ + } - {u_ - }} \right\vert$ and $ {\left\Vert {\left( {{v_0} - {v_ + }, {u_0} - {u_ + }} \right)} \right\Vert _{{H^1}}}$, nor restriction on $ \gamma \left( \ge 1 \right)$. A similar result to the corresponding Cauchy problem is also obtained. The proofs are given by an elementary $ {L^{2}}$-energy method.


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


Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35B40, 35L50, 76L05, 76N10

Retrieve articles in all journals with MSC: 35B40, 35L50, 76L05, 76N10


Additional Information

DOI: https://doi.org/10.1090/qam/1738558
Article copyright: © Copyright 2000 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website