Global Nonexistence of Smooth Electric Induction Fields in One-Dimensional Nonlinear Dielectrics.
SOUTH CAROLINA UNIV COLUMBIA DEPT OF MATHEMATICS COMPUTER SCIENCE AND STATISTICS
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Coupled nonlinear wave equations are derived for the evolution of the components of the electric induction field D in a class of rigid nonlinear dielectrics governed by the nonlinear constitutive relation E lambda DD, where E is the electric field and lambda greater than 0 is a scalar-valued vector function. For the special case of a finite one-dimensional dielectric rod, embedded in a perfect conductor, and subjected to an applied electric field, which is perpendicular to the axis of the rod, and depends only on variations of the coordinate along that axis, it is shown that, under relatively mild conditions on lambda, solutions of the corresponding initial-boundary value problem for the electric induction field can not exist globally in time in the L2 sense under slightly stronger assumptions on the constitutive function lambda, a standard Riemann Invariant argument may be applied to show that the space-time gradient of the non-zero component of the electric induction field must blow-up in finite time. Some growth estimates for solutions, which are valid on the maximal time-interval of existence are also derived.
- Electricity and Magnetism