Two-dimensional Riemann problem for a single conservation law

Source abstract

The entropy solutions to the partial differential equation $$( ∂ / ∂ t ) u ( t , x , y ) + ( ∂ / ∂ x ) f ( u ( t , x , y ) ) + ( ∂ / ∂ y ) g ( u ( t , x , y ) ) = 0 , (\partial /\partial t)u(t,x,y) + (\partial /\partial x)f(u(t,x,y)) + (\partial /\partial y)g(u(t,x,y)) = 0,$$ with initial data constant in each quadrant of the ( x , y ) (x,y) plane, have been constructed and are piecewise smooth under the condition f ( u ) ≠ 0 , g ( u ) ≠ 0 , ( f ( u ) / g ( u ) ) ′ ≠ 0 f(u) \ne 0, g(u) \ne 0, (f(u)/g(u))\prime \ne 0 . This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here.