TY - RPRT
T1 - Characteristic boundary layers for mixed hyperbolic systems in one space dimension and applications to the Navier-Stokes and MHD equations
Y1 - 2018
A1 - Stefano Bianchini
A1 - Laura Spinolo
AB - 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.
PB - SISSA
UR - http://preprints.sissa.it/handle/1963/35325
U1 - 35635
U2 - Mathematics
U4 - 1
ER -
TY - JOUR
T1 - Invariant manifolds for a singular ordinary differential equation
JF - Journal of Differential Equations 250 (2011) 1788-1827
Y1 - 2011
A1 - Stefano Bianchini
A1 - Laura Spinolo
PB - Elsevier
UR - http://hdl.handle.net/1963/2554
U1 - 1565
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - The boundary Riemann solver coming from the real vanishing viscosity approximation
JF - Arch. Ration. Mech. Anal. 191 (2009) 1-96
Y1 - 2009
A1 - Stefano Bianchini
A1 - Laura Spinolo
AB - 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.
UR - http://hdl.handle.net/1963/1831
U1 - 2385
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - A connection between viscous profiles and singular ODEs
JF - Rend. Istit. Mat. Univ. Trieste 41 (2009) 35-41
Y1 - 2009
A1 - Stefano Bianchini
A1 - Laura Spinolo
UR - http://hdl.handle.net/1963/2555
U1 - 1564
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - RPRT
T1 - Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems
Y1 - 2008
A1 - Stefano Bianchini
A1 - Laura Spinolo
UR - http://hdl.handle.net/1963/3400
U1 - 932
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -