# American Institute of Mathematical Sciences

July  2002, 8(3): 605-625. doi: 10.3934/dcds.2002.8.606

## The residual boundary conditions coming from the real vanishing viscosity method

 1 Unité de Mathématiques Pures et Appliquées, Ecole Normale Supérieure de Lyon, 46 Allée d'Italie, 69364 Lyon cedex 07, France

Received  June 2001 Revised  November 2001 Published  April 2002

We consider an initial boundary value problem set in $\{x > 0\}$ for a mixed hyperbolic parabolic system of conservation laws with a small parameter $\varepsilon$, $u_t +F(u)_x =\varepsilon(B(u)u_x)_x$. In the non-characteristic case a boundary layers analysis gives a set of boundary conditions, the set of residual boundary conditions $\mathcal C$, for the inviscid system $u_t+F(u)_x = 0$. We generalize the results of [16] obtained for strictly parabolic perturbations to a realistic setting. We show that the set $\mathcal C$ has the suitable geometric property to construct a solution of the inviscid sytem in a vicinity of a point where the Evans function of the corresponding profile of boundary layer does not vanish at zero. Next we consider multidimensional systems. We show that the Kreiss-Lopatinski determinant for the hyperbolic system linearized about a constant state in $\mathcal C$ is equal to the reduced Evans function for the viscous system linearized about the corresponding profile of boundary layer.
Citation: Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606
 [1] Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59 [2] Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1 [3] M.J. Lopez-Herrero. The existence of weak solutions for a general class of mixed boundary value problems. Conference Publications, 2011, 2011 (Special) : 1015-1024. doi: 10.3934/proc.2011.2011.1015 [4] Santiago Cano-Casanova. Coercivity of elliptic mixed boundary value problems in annulus of $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3819-3839. doi: 10.3934/dcds.2012.32.3819 [5] Aimin Huang, Roger Temam. The linear hyperbolic initial and boundary value problems in a domain with corners. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1627-1665. doi: 10.3934/dcdsb.2014.19.1627 [6] Martn P. Árciga Alejandre, Elena I. Kaikina. Mixed initial-boundary value problem for Ott-Sudan-Ostrovskiy equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 381-409. doi: 10.3934/dcds.2012.32.381 [7] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135 [8] Gung-Min Gie, Makram Hamouda, Roger Témam. Boundary layers in smooth curvilinear domains: Parabolic problems. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1213-1240. doi: 10.3934/dcds.2010.26.1213 [9] Vladimir V. Varlamov. On the initial boundary value problem for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 431-444. doi: 10.3934/dcds.1998.4.431 [10] Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271 [11] John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273 [12] Antonella Marini, Thomas H. Otway. Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric. Conference Publications, 2015, 2015 (special) : 801-808. doi: 10.3934/proc.2015.0801 [13] V. A. Dougalis, D. E. Mitsotakis, J.-C. Saut. On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1191-1204. doi: 10.3934/dcds.2009.23.1191 [14] Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046 [15] Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244 [16] Elena Rossi. Well-posedness of general 1D initial boundary value problems for scalar balance laws. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3577-3608. doi: 10.3934/dcds.2019147 [17] Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas, Kun Zhao. Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154 [18] Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001 [19] Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293 [20] Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51

2019 Impact Factor: 1.338