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Fractional Laplacian with singular drift

100%

Jakubowski T.

Studia Mathematica

2011

tom 207

nr 3

257-273

For α ∈ (1,2) we consider the equation $∂_t u = Δ^{α/2}u + b·∇u$, where b is a time-independent, divergence-free singular vector field of the Morrey class $M₁^{1-α}$. We show that if the Morrey norm $||b||_{M₁^{1-α}}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

Time-dependent Schrödinger perturbations of transition densities

51%

Bogdan K., Hansen W., Jakubowski T.

Studia Mathematica

2008

tom 189

nr 3

235-254

We construct the fundamental solution of $∂_t - Δ_y - q(t,y)$ for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We also discuss specific applications to Schrödinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator.

Ograniczanie wyników

2 Studia Mathematica

2 Jakubowski T.

1 Bogdan K.

1 Hansen W.

1 2011

1 2008

Wyniki wyszukiwania

Fractional Laplacian with singular drift

Time-dependent Schrödinger perturbations of transition densities