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fa.purposeful evaluation – Continuity of the entropy of the answer of a parabolic PDE at $t=0$ Answer

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fa.purposeful evaluation – Continuity of the entropy of the answer of a parabolic PDE at $t=0$

Consider the next preliminary worth drawback for a parabolic PDE:
$$ commence {circumstances} textrm {div} (A , nabla u (t, x)) , + , b (x) cdot nabla u (t, x) , = , partial_t u (t, x) quad x in mathbb R ^ d ,, t> 0 [4pt] u (0, x) , = , v (x) aim {circumstances} $$
Where $ A $ is a ceaseless positive-definite matrix, $ b $ is a flush vector bailiwick, $ v $ is a apt initiate date with
$$ v> 0 ,, int _ { mathbb R ^ d} v (x) , dx = 1 ,, int _ { mathbb R ^ d} v (x) , | log v (x) | , dx , < infty ,. $$
I’m within the probabilistic answer $ u $ of the issue, ie $ u> 0 ,, int _ { mathbb R ^ d} u (t, x) , dx = 1 $ for all $ t> 0 , $. I do know it exists and is exclusive and I do know it too
$$ int _ { mathbb R ^ d} | b (x) | ^ 2 , u (t, x) , dx , < infty $$
for all $ t geq0 , $. I wish to discover circumstances for the initiate date $ v $ they assure the Continuity of entropy $ t = 0 $, ie,
$$ int _ { mathbb R ^ d} u (t, x) , log u (t, x) , dx , xrightarrow[tto0]{} , int _ { mathbb R ^ d} v (x) , log v (x) , dx ,. $$
I solely create benchmark outcomes that assure
$$ int _ { mathbb R ^ d} | u (t, x) -v (x) | , dx , xrightarrow[tto0]{} , 0 $$
which seems to breathe a weaker situation.

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