Pointwise constraints for scalar conservation laws with positive wave velocity

Abstract We consider a conservation law with strictly positive wave velocity and study the well-posedness of a suitable notion of solution for the associated initial value problem under a flux constraint active in the half-line $${\mathbb {R}}_+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> . The strict positivity of the wave velocity allows for the dynamics in the unconstrained region $${\mathbb {R}}_-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:math> to be fully determined by the restriction of the initial data to $${\mathbb {R}}_-$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:math> . On the other hand, the solution in the constrained region is dictated by the assumption that the total mass of the initial datum is conserved along the evolution. We formulate the transmission condition at the interface $$\{x=0\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> in such a way that the boundary datum for the initial boundary value problem posed on $${\mathbb {R}}_+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> is given by the largest incoming flux that is admissible under the constraint, while the exceeding mass is accumulated in a “buffer” (as an atomic measure concentrated at the interface).