ABSTRACT
When an interface or free-boundary propagates normal to itself at a speed that depends non-monotonically on curvature alone, an anti-diffusive form of instability appears in the shape of the interface wherever the speed-curvature relationship has negative slope. If this slope is restricted to windows of curvature, then a remarkable degree of regularisation appears. An interface can be subjected to the anti-diffusive instability for an indefinite time but remain well-posed with respect to initial conditions for all time. This result is connected with the theorem: a diffusion equation ut = D▽2u, having time-dependent diffusion coefficient D(t), is ill-posed with respect to initial conditions when the integral of the diffusion coefficient becomes negative, which is not necessarily the same as the time when the diffusion coefficient itself becomes negative. On the other hand, the interface is structurally ill-posed when subject to the anti-diffusive instability, as described in [1]. With a non-monotonic speed curvature relationship, discontinuities in curvature can usually be expected to arise, rendering inappropriate some otherwise elegant results based on assuming continuity of curvature. These discontinuities separate curvature into ranges which appear to have much in common with separation into phases in material systems, although the separation of an interface into different “phases” of curvature involves important additional features. Some further discussion of the nature and consequence of non-monotonicity in the speed-curvature relationship is given in [1].
When an interface or free-boundary propagates normal to itself at a speed that depends non-monotonically on curvature alone, an anti-diffusive form of instability appears in the shape of the interface wherever the speed-curvature relationship has negative slope. If this slope is restricted to windows of curvature, then a remarkable degree of regularisation appears. An interface can be subjected to the anti-diffusive instability for an indefinite time but remain well-posed with respect to initial conditions for all time. This result is connected with the theorem: a diffusion equation ut = D▽2u, having time-dependent diffusion coefficient D(t), is ill-posed with respect to initial conditions when the integral of the diffusion coefficient becomes negative, which is not necessarily the same as the time when the diffusion coefficient itself becomes negative. On the other hand, the interface is structurally ill-posed when subject to the anti-diffusive instability, as described in [1]. With a non-monotonic speed curvature relationship, discontinuities in curvature can usually be expected to arise, rendering inappropriate some otherwise elegant results based on assuming continuity of curvature. These discontinuities separate curvature into ranges which appear to have much in common with separation into phases in material systems, although the separation of an interface into different “phases” of curvature involves important additional features. Some further discussion of the nature and consequence of non-monotonicity in the speed-curvature relationship is given in [1].
When an interface or free-boundary propagates normal to itself at a speed that depends non-monotonically on curvature alone, an anti-diffusive form of instability appears in the shape of the interface wherever the speed-curvature relationship has negative slope. If this slope is restricted to windows of curvature, then a remarkable degree of regularisation appears. An interface can be subjected to the anti-diffusive instability for an indefinite time but remain well-posed with respect to initial conditions for all time. This result is connected with the theorem: a diffusion equation ut = D▽2u, having time-dependent diffusion coefficient D(t), is ill-posed with respect to initial conditions when the integral of the diffusion coefficient becomes negative, which is not necessarily the same as the time when the diffusion coefficient itself becomes negative. On the other hand, the interface is structurally ill-posed when subject to the anti-diffusive instability, as described in [1]. With a non-monotonic speed curvature relationship, discontinuities in curvature can usually be expected to arise, rendering inappropriate some otherwise elegant results based on assuming continuity of curvature. These discontinuities separate curvature into ranges which appear to have much in common with separation into phases in material systems, although the separation of an interface into different “phases” of curvature involves important additional features. Some further discussion of the nature and consequence of non-monotonicity in the speed-curvature relationship is given in [1].
