scholarly journals Scattering theory in homogeneous Sobolev spaces for Schrödinger and wave equations with rough potentials

2020 ◽  
Vol 61 (9) ◽  
pp. 091505
Author(s):  
Haruya Mizutani
Keyword(s):  
Sobolev Spaces ◽  
Scattering Theory ◽  
Wave Equations ◽  
Homogeneous Sobolev Spaces

scholarly journals Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

2021 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed
Keyword(s):  
Rigid Body ◽  
Sobolev Spaces ◽  
Periodic Motion ◽  
Viscous Incompressible Fluid ◽  
Body Motion ◽  
Translational Velocity ◽  
Ill Posed ◽  
Homogeneous Sobolev Spaces ◽  
The Mean ◽  
Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

Bilinear forms on non-homogeneous Sobolev spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 995-1026
Author(s):  
Carme Cascante ◽  
Joaquín M. Ortega
Keyword(s):  
Sobolev Spaces ◽  
Bilinear Form ◽  
Positive Measure ◽  
Bilinear Forms ◽  
Homogeneous Sobolev Spaces

AbstractIn this paper, we show that if {b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, {0<s<1}, by(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x}is continuous if and only if the positive measure {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for {H_{s}^{2}(\mathbb{R}^{n})}.

scholarly journals Bilinear forms on homogeneous Sobolev spaces

2018 ◽  
Vol 457 (1) ◽  
pp. 722-750
Author(s):  
Carme Cascante ◽  
Joan Fàbrega ◽  
Joaquín M. Ortega
Keyword(s):  
Sobolev Spaces ◽  
Bilinear Forms ◽  
Homogeneous Sobolev Spaces

On Some Functional Equations of Carleman

1972 ◽  
Vol 15 (1) ◽  
pp. 129-131
Author(s):  
Charles G. Costley
Keyword(s):  
Scattering Theory ◽  
Functional Equations ◽  
Quantum Physics ◽  
Wave Equations ◽  
Hermitian Forms ◽  
Carleman Kernel ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations ◽  
Condition B

The celebrated Fredholm theory of linear integral equations holds if the kernel K(x, y) or one of its iterates K(n) is bounded. Hilbert utilizing his theory of quadratic form was able to extend the theory to the kernels K(x, y) satisfyingabwhere k is independent of u(x).These theories were extended considerably by T. Carleman who deleted condition (b) above.Equations involving this Carleman kernel have been found useful in connection with Hermitian forms, continued fractions, Schroedinger wave equations (see [1], [2]) and more recently in scattering theory in quantum physics, etc. [3]. See also [5] for a variety of applications and extensions.

scholarly journals Fractional Laplacian, homogeneous Sobolev spaces and their realizations

2020 ◽  
Vol 199 (6) ◽  
pp. 2243-2261 ◽  
Author(s):  
Alessandro Monguzzi ◽  
Marco M. Peloso ◽  
Maura Salvatori
Keyword(s):  
Sobolev Spaces ◽  
Fractional Laplacian ◽  
Homogeneous Sobolev Spaces
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