• Corpus ID: 230433741

On the heat equation with drift in $L_{d+1}$

  title={On the heat equation with drift in \$L\_\{d+1\}\$},
  author={Nicolai V. Krylov},
in the class of functions u ∈ ⋃T>0W 1,2 p ((−T, 1) × Rd) such that u = 0 for t = 1, where p < d+ 1, q is large enough, ∂t = ∂ ∂t , Di = ∂ ∂xi . A somewhat unusual feature of this problem is that bDiu 6∈ Lp((0, 1)×Rd) for arbitrary u ∈ W 1,2 p ((0, 1) × Rd) even vanishing for t = 1. Therefore, if we solve (1.2) and plug the solution into an equation with different b of the same class, we will generally not obtain a function in Lq even locally. The author is aware of only three similar occasions… 
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  • Partial Differential Equations, Vol
  • 1996