Asymptotic Behavior of Solutions of Certain Parabolic Problems with Space and Time Dependent Coefficients,
OREGON STATE UNIV CORVALLIS
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to complete the anti-dynamo theorem, the decay of the toroidal part of the magnetic field has to be proved. For those equations there is no maximum principle. But the proof for incompressible flows could be altered to cover the compressible case as well. Again, the techniques used were sufficient to prove the decay of the bounding function. But intuition and all numerical experiments suggested the stronger result that the solutions tend to zero. To prove this, much more involved mathematical techniques seemed to be necessary. The Liapunov function techniques were combined with the method of shrinking cylinders and a Harnack inequality to prove that the solutions tend to zero exponentially. The present paper applies methods to prove that the flux function tends to a constant exponentially. The Liapunov function part of the proof needs an auxiliary problem, whose solutions stay bounded for all times. Since this is a backward parabolic equation we also deal with its forward associate.
- Theoretical Mathematics
- Fluid Mechanics