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| For degenerate parabolic equations, the entropy dissipation may vanish for states other than the equilibrium. Hence, the aproach due to Bakry-Emery does not carry over.
In the hypocoercive case, we first establish a condition that is equivalent to the existence of a unique normalised steady state. By introducing an auxiliary functional we prove the exponential decay of the solution towards the steady state in relative entropy. Finally, we show that the obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). |
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