Scalars convected by a two‐dimensional incompressible flow

We provide a test for numerical simulations, for several two dimensional incompressible flows, that appear to develop sharp fronts. We show that in order to have a front the velocity has to have uncontrolled velocity growth. © 2001 John Wiley & Sons, Inc.

[1]  F. Smith,et al.  The formation of magnetic singularities by time-dependent collapse of an X -type magnetic field , 1995, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[2]  Michio Yamada,et al.  Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow , 1997 .

[3]  Diego Cordoba,et al.  Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation , 1998, math/9811184.

[4]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[5]  Alain Pumir,et al.  Development of singular solutions to the axisymmetric Euler equations , 1992 .

[6]  Rainer Grauer,et al.  Adaptive Mesh Refinement for Singular Current Sheets in Incompressible Magnetohydrodynamic Flows , 1997 .

[7]  Christiane Marliani,et al.  Evolution of current sheets and regularity of ideal incompressible magnetic fluids in 2D , 2000 .

[8]  E Weinan,et al.  Small‐scale structures in Boussinesq convection , 1998 .

[9]  Norbert Schorghofer,et al.  Nonsingular surface quasi-geostrophic flow , 1998, math/9805027.

[10]  Russel E. Caflisch,et al.  Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD , 1997 .

[11]  Eugene N. Parker,et al.  Spontaneous current sheets in magnetic fields : with applications to stellar x-rays , 1994 .

[12]  Andrew J. Majda,et al.  Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar , 1994 .

[13]  C. Fefferman,et al.  Geometric constraints on potentially singular solutions for the 3-D Euler equations , 1996 .