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Abstract In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane.We consider equations of the form Δ u + W ⋅ ∇ u = 0 {\Delta u+W\cdot\nabla u=0} in ℝ 2 {\mathbb{R}^{2}} ,where W = W 1 + i W 2 {W=W_{1}+iW_{2}} with each W j {W_{j}} being real-valued.Under the assumptions that W j ∈ L q j {W_{j}\in L^{q_{j}}} for some q 1 ∈ [ 2 , ∞ ] {q_{1}\in[2,\infty]} , q 2 ∈ ( 2 , ∞ ] {q_{2}\in(2,\infty]} and that W 2 {W_{2}} exhibits rapid decay at infinity,we prove new global unique continuation estimates.This improvement is accomplished by reducing our equations to vector-valued Beltrami systems.Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.
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Global Sobolev persistence for the fractional Boussinesq equation with zero diffusivity
We prove the persistence of regularity for the 2D alpha-franctional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e. for (u_0,rho_0)\in W^s,q(R^2)\times W^s,q(R^2), where s>1 and 2
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- Award ID(s):
- 1907992
- PAR ID:
- 10173655
- Date Published:
- Journal Name:
- Pure and applied functional analysis
- Volume:
- 5
- Issue:
- 1
- ISSN:
- 2189-3756
- Page Range / eLocation ID:
- 27-45
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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