Non-Linear Damping of Surface Alfvén Waves Due to Uniturbulence

This investigation is concerned with uniturbulence associated with surface Alfvén waves that exist in a Cartesian equilibrium model with a constant magnetic field and a piece-wise constant density. The surface where the equilibrium density changes in a discontinuous manner are the source of surface Alfvén waves. These surface Alfvén waves create uniturbulence because of the variation of the density across the background magnetic field. The damping of the surface Alfvén waves due to uniturbulence is determined using the Elsässer formulation. Analytical expressions for the wave energy density, the energy cascade, and the damping time are derived. The study of uniturbulence due to surface Alfvén waves is inspired by the observation that (the fundamental radial mode of) kink waves behave similarly to surface Alfvén waves. The results for this relatively simple case of surface Alfvén waves can help us understand the more complicated case of kink waves in cylinders. We perform a series of 3D ideal MHD simulations for a numerical demonstration of the non-linearly self-cascading model of unidirectional surface Alfvén waves using the code MPI-AMRVAC. We show that surface Alfvén waves damping time in the numerical simulations follows well our analytical prediction for that quantity. Analytical theory and the simulations show that the damping time is inversely proportional to the amplitude of the surface Alfvén waves and the density contrast. This unidirectional cascade may play a role in heating the coronal plasma.

Lp-asymptotic stability of 1D damped wave equations with localized and linear damping

<p>In this paper, we study the L^p-asymptotic stability of the one dimensional linear damped<br />wave equation with Dirichlet boundary conditions in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8712;</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>&#8734;</mo><mo>)</mo></math>. The damping<br />term is assumed to be linear and localized&nbsp; to an arbitrary open sub-interval of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>. We prove that the&nbsp;<br />semi-group <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>p</mi></msub><mo>(</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>t</mi><mo>&#8805;</mo><mn>0</mn></mrow></msub></math> associated with the previous equation is well-posed and exponentially stable.<br />The proof relies on the multiplier method and depends on whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8805;</mo><mn>2</mn></math>&nbsp;or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>&#60;</mo><mi>p</mi><mo>&#60;</mo><mn>2</mn></math>.</p>

Nonexistence of Global Solutions of Systems of Time Fractional Differential equations posed on the Heisenberg group

We first consider the nonlinear time fractional diffusion equation D^{1+α}u+D^β u−∆_{H} u=|u|^p posed on the Heisenberg group H, where 1 < p is a positive real nimber to be specified later; D^δ_{0|t} is the Liouville-Caputo derivative of order δ. For 0 < α < 1,0 < β ≤ 1. This equation interpolates the heat equation and the wave equation with the linear damping D^β_{0|t}u. We present the Fujita exponent for blow-up. Then establish sufficient conditions ensuring non-existence of local solutions. We extend the analysis to the case of the system D^{1+α}u+D^β u−∆_{H} u=|v|^q D^{1+δ}v+D^γ v−∆_{H} v=|u|^p. Our method of proof is based on the nonlinear capacity method.

Modeling the Dynamics of a Gyroscopic Rigid Rotor with Linear and Nonlinear Damping and Nonlinear Stiffness of the Elastic Support

This study analytically and numerically modeled the dynamics of a gyroscopic rigid rotor with linear and nonlinear cubic damping and nonlinear cubic stiffness of an elastic support. It has been shown that (i) joint linear and nonlinear cubic damping significantly suppresses the vibration amplitude (including the maximum) in the resonant velocity region and beyond it, and (ii) joint linear and nonlinear cubic damping more effectively affects the boundaries of the bistability region by its narrowing than linear damping. A methodology is proposed for determining and identifying the coefficients of nonlinear stiffness, linear damping, and nonlinear cubic damping of the support material, where jump-like effects are eliminated. Damping also affects the stability of motion; if linear damping shifts the left boundary of the instability region towards large amplitudes and speeds of rotation of the shaft, then nonlinear cubic damping can completely eliminate it. The varying amplitude (VAM) method is used to determine the nature of the system response, supplemented with the concept of “slow” time, which allows us to investigate and analyze the effect of nonlinear cubic damping and nonlinear rigidity of cubic order on the frequency response at a nonstationary resonant transition.