• Corpus ID: 119343088

Scalars convected by a 2D incompressible flow

@article{Crdoba2001ScalarsCB,
  title={Scalars convected by a 2D incompressible flow},
  author={Diego C{\'o}rdoba and Charles Fefferman},
  journal={arXiv: Analysis of PDEs},
  year={2001}
}
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. 
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References

SHOWING 1-6 OF 6 REFERENCES
Adaptive Mesh Refinement for Singular Current Sheets in Incompressible Magnetohydrodynamic Flows
The formation of current sheets in ideal incompressible magnetohydrodynamic flows in two dimensions is studied numerically using the technique of adaptive mesh refinement. The growth of current
Remarks on the breakdown of smooth solutions for the 3-D Euler equations
The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially
Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow
The growth of the gradient of a scalar temperature in a quasigeostrophic flow is studied numerically in detail. We use a flow evolving from a simple initial condition which was regarded by Constantin
Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar
The formation of strong and potentially singular fronts in a two-dimensional quasigeostrophic active scalar is studied through the symbiotic interaction of mathematical theory and numerical
Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD
Abstract: For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B3s with s greater than 1/3. B3s consists of functions that are