Occurrence Of Finite-time Singularities Research Articles

On the occurrence of finite-time singularities in swampland-related quintessence dark energy models

In this work, we focus on the phase space singularities of interactive quintessence model in the presence of matter fluid. This model is related to swampland studies, that the outcomes affect all these swampland-related models with the same dynamical system. We shall form the dynamical system corresponding to the cosmological system, which is eventually autonomous, and by using the dominant balances technique we shall investigate the occurrence or not of finite-time singularities. Our results indicate that the dynamical system of the model may develop finite-time singularities, but these are not general singularities, like in the case that the matter fluids were absent, in which case singularities occurred for general initial conditions. Hence, the presence of matter fluids affects the dynamical system of the cosmological system, making the singularities to depend on the initial conditions, instead of occurring for general initial conditions.

Singularity Formation in Fractional Burgers’ Equations

The formation of singularities in finite time in non-local Burgers' equations, with time-fractional derivative, is studied in detail. The occurrence of finite time singularity is proved, revealing the underlying mechanism, and precise estimates on the blow-up time are provided. The employment of the present equation to model a problem arising in job market is also analyzed.

A survey on second order free boundary value problems modelling MEMS with general permittivity profile

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles. The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

A fourth-order model for MEMS with clamped boundary conditions

The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter $\lambda$. It is shown that there is a threshold value $\lambda_*>0$ of the voltage parameter such that no radially symmetric stationary solution exists for $\lambda>\lambda_*$, while at least two such solutions exist for $\lambda\in (0,\lambda_*)$. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $\lambda>\lambda_*$.

A free boundary problem modeling electrostatic MEMS: I. Linear bending effects

The dynamical and stationary behaviors of a fourth-order evolution equation with clamped boundary conditions and a singular nonlocal reaction term, which is coupled to an elliptic free boundary problem in a non-smooth domain, are investigated. The equation arises in the modeling of microelectromechanical systems and includes two positive parameters $$\lambda $$ and $$\varepsilon $$ related to the applied voltage and the aspect ratio of the device, respectively. Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant. Existence of a stable steady state is shown for sufficiently small $$\lambda $$ . Non-existence of steady states is also established when $$\varepsilon $$ is small enough and $$\lambda $$ is large enough (depending on $$\varepsilon $$ ).

Spiky Development at the Interface in Rayleigh-Taylor Instability: Layzer Approximation with Second Harmonic

Layzer’s approximation method for investigation of two fluid interface structures associated with Rayleigh Taylor instability for arbitrary Atwood number is extended with the inclusion of second harmonic mode leaving out the zeroth harmonic one. The modification makes the fluid velocities vanish at infinity and leads to avoidance of the need to make the unphysical assumption of the existence of a time dependent source at infinity. The present analysis shows that for an initial interface perturbation with curvature exceeding , where A is the Atwood number there occurs an almost free fall of the spike with continuously increasing sharpening as it falls. The curvature at the tip of the spike also increases with Atwood number. Certain initial condition may also result in occurrence of finite time singularity as found in case of conformal mapping technique used earlier. However bubble growth rate is not appreciably affected.

Occurrence of finite-time singularities in epidemic models of rupture, earthquakes, and starquakes.

We present a new kind of critical stochastic finite-time singularity, relying on the interplay between long-memory and extreme fluctuations. We illustrate it on the well-established epidemic-type aftershock model for aftershocks, based solely on the most solidly documented stylized facts of seismicity (clustering in space and in time and power law Gutenberg-Richter distribution of earthquake energies). This theory accounts for the main observations (power law acceleration and discrete scale invariant structure) of critical rupture of heterogeneous materials, of the largest sequence of starquakes ever attributed to a neutron star, as well as of earthquake sequences.