On the Inverse Source Identification Problem in $L^{\infty }$ for Fully Nonlinear Elliptic PDE

In this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the $L^{\infty }$ L ∞ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.

Asymptotic derivation and simulations of a non-local Exner model in large viscosity regime

The present paper deals with the modeling and numerical approximation of bed load transport under the action of water. A new shallow water type model is derived from the stratified two-fluid Navier-Stokes equations. Its novelty lies in the magnitude of a viscosity term that leads to a momentum equation of elliptic type. The full model, sediment and water, verifies a dissipative energy balance for smooth solutions. The numerical resolution of the sediment layer is not trivial since the viscosity introduces a non-local term in the model. Adding a transport threshold makes the resolution even more challenging. A schema based on a staggered discretization is proposed for the full model, sediment and water.

BUBBLE DYNAMICS-BASED MODELING OF THE CAVITATION DYNAMICS IN LUBRICATED CONTACTS

Cavitation is a common phenomenon in fluid machinery and lubricated contacts. In lubricated contacts, there is a presumption that the short-term tensile stresses at the onset of bubble formation have an influence on material wear. To investigate the duration and magnitude of tensile stresses in lubricating films using numerical simulation, a suitable simulation model must be developed. The chosen simulation approach with bubble dynamics is based on the coupling of the Reynolds equation and Rayleigh-Plesset equation (introduced about 20 years ago by Someya).Following the basic approach from the author’s earlier papers on the negative squeeze motion with bubble dynamics for the simulation of mixed lubrication of rough surfaces, the paper at hand shows modifications to the Rayleigh-Plesset equation that are required to get the time scale for the dynamic processes right. This additional term is called the dilatational viscosity term, and it significantly influences the behavior of the numerical model.

Continuous solution for a non-linear eikonal system

<p style='text-indent:20px;'>In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. We prove a global existence result in the framework of continuous viscosity solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity, relying on a new gradient entropy and <inline-formula><tex-math id="M1">\begin{document}$ BV $\end{document}</tex-math></inline-formula> estimates. A uniqueness result is also proved through a comparison principle property.</p>

Antisolar differential rotation of slowly rotating cool stars

Rotating stellar convection transports angular momentum towards the equator, generating the characteristic equatorial acceleration of the solar rotation while the radial flux of angular momentum is always inwards. New numerical box simulations for the meridional cross-correlation ⟨uθuϕ⟩, however, reveal the angular momentum transport towards the poles for slow rotation and towards the equator for fast rotation. The explanation is that for slow rotation a negative radial gradient of the angular velocity always appears, which in combination with a so-far neglected rotation-induced off-diagonal eddy viscosity term ν⊥ provides “antisolar rotation” laws with a decelerated equator. Similarly, the simulations provided positive values for the rotation-induced correlation ⟨uruθ⟩, which is relevant for the resulting latitudinal temperature profiles (cool or warm poles) for slow rotation and negative values for fast rotation. Observations of the differential rotation of slowly rotating stars will therefore lead to a better understanding of the actual stress-strain relation, the heat transport, and the underlying model of the rotating convection.

Study of Enhanced Self Mixing in Ferrofluid Flow in an Elbow Channel Under the Influence of Non-Uniform Magnetic Field

This paper investigates numerically the various parameters dictating the vortical (self)-mixing induced by a non-uniform magnetic field in a ferrofluid flow in an elbow channel. The elbow bend region of the channel has two current carrying conductors placed symmetrically and parametrically from the channel and are used to generate a non-uniform magnetic field. The ferrofluid is assumed to be pre-magnetized, isothermal and electrically non-conductive as it enters the channel and has a prescribed inlet magnetization and temperature. The mixing efficiency is characterized by introducing different mixing scalars based on velocity of the fluid and are compared in order to determine the overall suitability of each scalar to quantify the flow vortical (self)-mixing. Parametric studies were performed by varying parameters influencing the magnetic field and the initial flow field. This resulted in variations in non-dimensional groups which control different aspects of the flow and helped establish their relationship with mixing efficiency. It was found that at higher Reynolds numbers the flow mixing induced by the lateral gradient in the Kelvin Body Force (KBF) dissipates and higher electrical inputs are required to sustain mixing in the flow. The effects of mixing enhancement on the pressure gradient across the channel was also established, along with the introduction of an enhanced viscosity term which is due to the non-collinearity of the magnetization vector and the magnetic field vector.

A bulk-interface correspondence for equatorial waves

Topology is introducing new tools for the study of fluid waves. The existence of unidirectional Yanai and Kelvin equatorial waves has been related to a topological invariant, the Chern number, that describes the winding of $f$-plane shallow water eigenmodes around band-crossing points in parameter space. In this previous study, the topological invariant was a property of the interface between two hemispheres. Here we ask whether a topological index can be assigned to each hemisphere. We show that this can be done if the shallow water model in the $f$-plane geometry is regularized by an additional odd-viscosity term. We then compute the spectrum of a shallow water model with a sharp equator separating two flat hemispheres, and recover the Kelvin and Yanai waves as two exponentially trapped waves along the equator, with all the other modes delocalized into the bulk. This model provides an exactly solvable example of bulk-interface correspondence in a flow with a sharp interface, and offers a topological interpretation for some of the transition modes described by Iga (J. Fluid Mech., vol. 294, 1995, pp. 367–390). It also paves the way towards a topological interpretation of coastal Kelvin waves along a boundary and, more generally, to an understanding of bulk-boundary correspondence in continuous media.