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 - On Bressan\\\'s conjecture on mixing properties of vector fields
JF - Self-Similar Solutions of Nonlinear PDE / Ed. Piotr Biler and Grzegorz Karch. - Banach Center Publ. 74 (2006) 13-31
Y1 - 2006
A1 - Stefano Bianchini
UR - http://hdl.handle.net/1963/1806
U1 - 2408
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - BV solutions for a class of viscous hyperbolic systems
JF - Indiana Univ. Math. J. 49 (2000) 1673-1714
Y1 - 2000
A1 - Stefano Bianchini
A1 - Alberto Bressan
PB - Indiana University Mathematics Journal
UR - http://hdl.handle.net/1963/3194
U1 - 1107
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -