Uniqueness Domains for $$\textbf{L}^\infty $$ Solutions of $$2\times 2$$ Hyperbolic Conservation Laws
Year: 2025
Type: article
Abstract: For a genuinely nonlinear $$2\times 2$$ 2 × 2 hyperbolic system of conservation laws, assuming that the initial data have a small $$\textbf{L}^\infty $$ L ∞ norm but a possibly unbounded total variation, the existence of global solutions was proven in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like $$t^{-1}$$ t - 1 . Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with a faster decay rate: $$\hbox {Tot.Var.}\bigl \{u(t,\cdot )\bigr \}\le C t^{\alpha -1}$$ Tot.Var. { u ( t , · ) } ≤ C t α - 1 . For these solutions, a uniqueness theorem is proven. Indeed, as the initial data range over a domain of functions with $$\Vert {\bar{u}}\Vert _{\textbf{L}^\infty } \le \varepsilon _1$$ ‖ u ¯ ‖ L ∞ ≤ ε 1 small enough, solutions with a fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to 1 by further shrinking the value of $$\varepsilon _1>0$$ ε 1 > 0 . An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.
Authors: Alberto Bressan, Elio Marconi, Ganesh Vaidya
Institutions: Pennsylvania State University, University of Padua
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4416040076