A variational formulation for constrained quasilinear vector systems

Abstract

Abstract.A variational formulation for multi-dimensional initial-and/or boundaryvalue problems for a system of quasilinear conservation equations with a rotationality condition in a vector form with the aid of a vector Lagrange multiplier is given. The duality between the physical and 'phase' (or hodograph) spaces emerges, and the Lagrange multiplier turns out to be the vector potential for the conserved field, and hence of some interest in itself. Application is given to a family of transonic flows in the physical and hodograph planes, and to a problem in nonlinear sound propagation. Introduction.Variational formulations equivalent to boundary-value problems for ordinary and elliptic partial differential equations have been of common use in particle and continuum mechanics and other branches of physics. For non-elliptic equations (e.g. diffusion and wave equations) the Dirichlet problem is improperly posed, and different conditions on parts of the boundary only have to be prescribed. Variational formulations for initial (rather than boundary) value problems have been proposed for linear parabolic and hyperbolic equations [1][2][3][4][5].The two-dimensional nonlinear case was worked out in [6] with the aid of a Lagrange multiplier. It is extended here to three independent variables. The three-dimensional case is presented in vector notation which brings out the geometric and physical meaning of the various terms and the vector Lagrange multiplier, and is invariant under coordinate transformations. For more than three independent space variables, the constraints and

Alternative variational formulations for first order partial differential systems

“…The results (16) have been described and analyzed in [8,9] where applications to specific flows are given. While (16) requires that u be smoother than A, the opposite holds for (17), which is more in accord with the physics of the problem, where the field variables u are related to the derivatives of the ' vector potential' A.…”