@article {2018,
title = {Characteristic boundary layers for mixed hyperbolic systems in one space dimension and applications to the Navier-Stokes and MHD equations},
number = {SISSA;45/2018/MATE},
year = {2018},
institution = {SISSA},
abstract = {We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the
mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null conductivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to non-conservative systems.},
url = {http://preprints.sissa.it/handle/1963/35325},
author = {Stefano Bianchini and Laura Spinolo}
}
@article {2011,
title = {Invariant manifolds for a singular ordinary differential equation},
journal = {Journal of Differential Equations 250 (2011) 1788-1827},
number = {SISSA;04/2008/M},
year = {2011},
publisher = {Elsevier},
doi = {10.1016/j.jde.2010.11.010},
url = {http://hdl.handle.net/1963/2554},
author = {Stefano Bianchini and Laura Spinolo}
}
@article {2009,
title = {The boundary Riemann solver coming from the real vanishing viscosity approximation},
journal = {Arch. Ration. Mech. Anal. 191 (2009) 1-96},
number = {SISSA;24/2006/M},
year = {2009},
abstract = {We study the limit of the hyperbolic-parabolic approximation $$ \\\\begin{array}{lll} v_t + \\\\tilde{A} ( v, \\\\, \\\\varepsilon v_x ) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in R^N\\\\\\\\ \\\\tilde \\\\beta (v (t, \\\\, 0)) = \\\\bar g \\\\\\\\ v (0, \\\\, x) = \\\\bar v_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nThe function $\\\\tilde \\\\beta$ is defined in such a way to guarantee that the initial boundary value problem is well posed even if $\\\\tilde \\\\beta$ is not invertible.\\nThe data $\\\\bar g$ and $\\\\bar v_0$ are constant. When $\\\\tilde B$ is invertible, the previous problem takes the simpler form $$ \\\\left\\\\{ \\\\begin{array}{lll} v_t + \\\\tilde{A} \\\\big( v, \\\\, \\\\varepsilon v_x \\\\big) v_x = \\\\varepsilon \\\\tilde{B}(v ) v_{xx} \\\\qquad v \\\\in \\\\mathbb{R}^N\\\\\\\\ v (t, \\\\, 0) \\\\equiv \\\\bar v_b \\\\\\\\ v (0, \\\\, x) \\\\equiv \\\\bar{v}_0. \\\\\\\\ \\\\end{array} \\\\right. $$\\nAgain, the data $\\\\bar v_b$ and $\\\\bar v_0$ are constant. The conservative case is included in the previous formulations. It is assumed convergence of the v, smallness of the total variation and other technical hypotheses and it is provided a complete characterization of the limit. The most interesting points are the following two. First, the boundary characteristic case is considered, i.e. one eigenvalue of $\\\\tilde A$ can be 0.\\n Second, as pointed out before we take into account the possibility that $\\\\tilde B$ is not invertible. To deal with this case, we take as hypotheses conditions that were introduced by Kawashima and Shizuta relying on physically meaningful examples. We also introduce a new condition of block linear degeneracy. We prove that, if it is not satisfied, then pathological behaviours may occur.},
doi = {10.1007/s00205-008-0177-6},
url = {http://hdl.handle.net/1963/1831},
author = {Stefano Bianchini and Laura Spinolo}
}
@article {2009,
title = {A connection between viscous profiles and singular ODEs},
journal = {Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41},
number = {SISSA;05/2008/M},
year = {2009},
url = {http://hdl.handle.net/1963/2555},
author = {Stefano Bianchini and Laura Spinolo}
}
@article {2008,
title = {Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems},
number = {SISSA;83/2008/M},
year = {2008},
url = {http://hdl.handle.net/1963/3400},
author = {Stefano Bianchini and Laura Spinolo}
}