%0 Report
%D 2018
%T Characteristic boundary layers for mixed hyperbolic systems in one space dimension and applications to the Navier-Stokes and MHD equations
%A Stefano Bianchini
%A Laura Spinolo
%X 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.
%I SISSA
%G en
%U http://preprints.sissa.it/handle/1963/35325
%1 35635
%2 Mathematics
%4 1
%$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2018-10-16T11:48:42Z
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%0 Journal Article
%J Journal of Differential Equations 250 (2011) 1788-1827
%D 2011
%T Invariant manifolds for a singular ordinary differential equation
%A Stefano Bianchini
%A Laura Spinolo
%B Journal of Differential Equations 250 (2011) 1788-1827
%I Elsevier
%G en_US
%U http://hdl.handle.net/1963/2554
%1 1565
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-01-15T10:35:53Z\\r\\nNo. of bitstreams: 1\\r\\nSissa0408M.pdf: 296248 bytes, checksum: 7563bf5d03a78a0e2c2c606a9b609827 (MD5)
%R 10.1016/j.jde.2010.11.010
%0 Journal Article
%J Arch. Ration. Mech. Anal. 191 (2009) 1-96
%D 2009
%T The boundary Riemann solver coming from the real vanishing viscosity approximation
%A Stefano Bianchini
%A Laura Spinolo
%X 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.
%B Arch. Ration. Mech. Anal. 191 (2009) 1-96
%G en_US
%U http://hdl.handle.net/1963/1831
%1 2385
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-06-08T11:20:08Z\\nNo. of bitstreams: 1\\nmath.AP-0605575.pdf: 716141 bytes, checksum: 7686c7f41f7b1f82a595f668049640f2 (MD5)
%R 10.1007/s00205-008-0177-6
%0 Journal Article
%J Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41
%D 2009
%T A connection between viscous profiles and singular ODEs
%A Stefano Bianchini
%A Laura Spinolo
%B Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41
%G en_US
%U http://hdl.handle.net/1963/2555
%1 1564
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-01-15T10:38:45Z\\nNo. of bitstreams: 1\\nSissa0508M.pdf: 107079 bytes, checksum: 5af029f1d9bcd4f14be2d8774cfa251e (MD5)
%0 Report
%D 2008
%T Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems
%A Stefano Bianchini
%A Laura Spinolo
%G en_US
%U http://hdl.handle.net/1963/3400
%1 932
%2 Mathematics
%3 Functional Analysis and Applications
%$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-15T12:03:08Z\\nNo. of bitstreams: 1\\nBianchini_Spinolo.pdf: 146920 bytes, checksum: 27283c16ae26e782e9f0d17c1fbb29d4 (MD5)