A variational principle for the nonlinear heat equation. (Q1432230)

The author studies Lagrangians of the type \[ {\mathcal L}_{\pm}={\mathcal L}^{(1)}\pm{\mathcal L}^{(2)}\equiv\frac 12[(\partial_t\theta)^2\pm \chi^2(\Delta\theta)^2], \] in order to obtain two formulations of the variational principle for the nonlinear heat equation. It is first established that a direct variational description of parabolic dissipative nonlinear evolution equations of the first order in \(t\) is impossible. The main steps in the construction developed in the paper are the following: (i) the desired function is replaced by potentials of the first two orders in the coordinate and (ii) the solutions to the original equation expressed in terms of potentials are proved to be solutions to the corresponding equations of the second order in \(t\). The desired Lagrangians for the last equations are then constructed. The situations considered in the paper occur in the dynamics of an ideal fluid (where the Clebsch transformation is used to replace the velocity by three scalars) and in electromagnetics (where the field components are introduced by scalar and vector potentials).