 # 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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