scholarly journals Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

2018 ◽  
Vol 69 (2) ◽  
Author(s):  
Paolo Antonelli ◽  
Alessandro Michelangeli ◽  
Raffaele Scandone
Keyword(s):  
Weak Solutions ◽  
Finite Energy ◽  
Time Dependent ◽  
Magnetic Potentials

scholarly journals Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

2020 ◽  
Vol 40 (6) ◽  
pp. 725-736
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Existence And Uniqueness ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Energy ◽  
Time Dependent ◽  
Energy Solution ◽  
Noncylindrical Domain ◽  
Finite Energy Solution ◽  
Initial Boundary Value

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

scholarly journals A minimization approach to the wave equation on time-dependent domains

2020 ◽  
Vol 13 (4) ◽  
pp. 425-436 ◽  
Author(s):  
Gianni Dal Maso ◽  
Lucia De Luca
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Space Time ◽  
Time Dependent ◽  
Existence Of Weak Solutions ◽  
Homogeneous Wave ◽  
Minimization Approach ◽  
Homogeneous Wave Equation

AbstractWe prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

Analytical Solution for the Creep Problem of a Rotating Functionally Graded Magneto–Electro–Elastic Hollow Cylinder in Thermal Environment

2019 ◽  
Vol 11 (06) ◽  
pp. 1950053 ◽  
Author(s):  
M. Saadatfar
Keyword(s):  
Differential Equation ◽  
Hollow Cylinder ◽  
Thermal Loading ◽  
Thermal Environment ◽  
Functionally Graded ◽  
Time Dependent ◽  
Creep Stress ◽  
Creep Analysis ◽  
Magnetic Potentials ◽  
Creep Strains

In this paper, an analytical method is presented for the problem of the time-dependent response of a functionally graded magneto–electro–elastic (FGMEE) rotating hollow cylinder in thermal environment. The material properties of FGMEE are supposed to be power-law functions of radius. Applying the equations of equilibrium and electrostatic and magnetostatic equations, a differential equation which includes creep strains is achieved. At first, an exact solution for the primitive stresses, electric and magnetic potentials are achieved by eliminating creep strains in the mentioned differential equation. Then, Prandtl–Reuss equations, as well as Norton’s law, are employed for the FGMEE. Now, creep stress rates can be achieved by considering only creep strains in the mentioned differential equation. As a final step, time-dependent creep stress, electric potential and magnetic potential redistributions at any time can be achieved using an iterative method. Numerical examples are presented to disclose the influence of creep evolution, thermal loading, angular velocity and grading index on the primitive and creep response of the FGMEE hollow cylinder. Results show that the enhancement in tensile hoop stress during the creep evolution must be considered in designing. So, the creep analysis is vital to have more reliable and accurate aerospace smart structures.

scholarly journals A System of Differential Set-Valued Variational Inequalities in Finite Dimensional Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wei Li ◽  
Xing Wang ◽  
Nan-Jing Huang
Keyword(s):  
Variational Inequalities ◽  
Existence Theorem ◽  
Weak Solutions ◽  
Convergence Result ◽  
Time Dependent ◽  
Euclidean Spaces ◽  
Finite Dimensional ◽  
Differential Set

A system of differential set-valued variational inequalities is introduced and studied in finite dimensional Euclidean spaces. An existence theorem of weak solutions for the system of differential set-valued variational inequalities in the sense of Carathéodory is proved under some suitable conditions. Furthermore, a convergence result on Euler time-dependent procedure for solving the system of differential set-valued variational inequalities is also given.

TOPOLOGICAL EFFECTS IN QUANTUM MECHANICS AND HIGH VELOCITY ESTIMATES

2011 ◽  
Vol 20 (05) ◽  
pp. 951-961 ◽  
Author(s):  
RICARDO WEDER
Keyword(s):  
Exact Solution ◽  
Quantum Mechanics ◽  
Error Bounds ◽  
High Velocity ◽  
Wave Packets ◽  
Finite Energy ◽  
Time Dependent ◽  
Rigorous Proof ◽  
Bohm Effect ◽  
Aharonov Bohm

We consider the problem of obtaining high-velocity estimates for finite energy solutions (wave packets) to Schrödinger equations for N-body systems. We discuss a time-dependent method that allows us to obtain precise estimates with error bounds that decay as a power of the velocity. We apply this method to the electric Aharonov–Bohm effect. We give the first rigorous proof that quantum mechanics predicts the existence of this effect. Our result follows from an estimate in norm, uniform in time, that proves that the Aharonov–Bohm Ansatz is a good approximation to the exact solution to the Schrödinger equation for high velocity.

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