Uniqueness Domains for $$\textbf{L}^\infty $$ Solutions of $$2\times 2$$ Hyperbolic Conservation Laws

Abstract For a genuinely nonlinear $$2\times 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> hyperbolic system of conservation laws, assuming that the initial data have a small $$\textbf{L}^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> 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 $$t^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> . Motivated by the theory of fractional domains for linear analytic semigroups, we consider here solutions with a faster decay rate: $$\hbox {Tot.Var.}\bigl \{u(t,\cdot )\bigr \}\le C t^{\alpha -1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Tot.Var.</mml:mtext> <mml:mrow> <mml:mo>{</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mo>·</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . For these solutions, a uniqueness theorem is proven. Indeed, as the initial data range over a domain of functions with $$\Vert {\bar{u}}\Vert _{\textbf{L}^\infty } \le \varepsilon _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mover> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mo>‖</mml:mo> </mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:msub> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>ε</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> 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 $$\varepsilon _1&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ε</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation.