Strong time regularity and decay of $${\textbf{L}}^\infty $$ solutions to $$2\times 2$$ systems of conservation laws

Abstract We consider $$\textbf{L}^\infty $$ <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> solutions to $$2\times 2$$ <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> systems of conservation laws. For genuinely nonlinear systems we prove that finite entropy solutions (in particular entropy solutions, if a uniformly convex entropy exists) belong to $$C^0(\mathbb {R}^+,; \textbf{L}^1_{loc}(\mathbb {R}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>,</mml:mo> <mml:mo>;</mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>loc</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Our second result establishes a dispersive-type decay estimate for vanishing viscosity solutions. Both results are unified by the use of a kinetic formulation.