Approximation of 2D Euler Equations by the Second-Grade Fluid Equations with Dirichlet Boundary Conditions

The second-grade fluid equations are a model for viscoelastic fluids, with two parameters: $\alpha > 0$, corresponding to the elastic response, and $\nu > 0$, corresponding to viscosity. Formally setting these parameters to $0$ reduces the equations to the incompressible Euler equations of ideal fluid flow. In this article we study the limits $\alpha, \nu \to 0$ of solutions of the second-grade fluid system, in a smooth, bounded, two-dimensional domain with no-slip boundary conditions. This class of problems interpolates between the Euler-$\alpha$ model ($\nu = 0$), for which the authors recently proved convergence to the solution of the incompressible Euler equations, and the Navier-Stokes case ($\alpha = 0$), for which the vanishing viscosity limit is an important open problem. We prove three results. First, we establish convergence of the solutions of the second-grade model to those of the Euler equations provided $\nu = \mathcal{O}(\alpha^2)$, as $\alpha \to 0$, extending the main result in [19]. Second, we prove equivalence between convergence (of the second-grade fluid equations to the Euler equations) and vanishing of the energy dissipation in a suitably thin region near the boundary, in the asymptotic regime $\nu = \mathcal{O}(\alpha^{6/5})$, $\nu/\alpha^2 \to \infty$ as $\alpha \to 0$. This amounts to a convergence criterion similar to the well-known Kato criterion for the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations. Finally, we obtain an extension of Kato's classical criterion to the second-grade fluid model, valid if $\alpha = \mathcal{O}(\nu^{3/2})$, as $\nu \to 0$. The proof of all these results relies on energy estimates and boundary correctors, following the original idea by Kato.Comment: 20pages,1figur

Convergence of the 2D Euler-α\alpha to Euler equations in the Dirichlet case: indifference to boundary layers

In this article we consider the Euler-$\alpha$ system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler-$\alpha$ regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler-$\alpha$ system approximate, in a suitable sense, as the regularization parameter $\alpha \to 0$, the initial velocity for the limiting Euler system. For small values of $\alpha$, this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler-$\alpha$ system converge, as $\alpha \to 0$, to the corresponding solution of the Euler equations, in $L^2$ in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the $\alpha \to 0$ limit, which underlies our work.Comment: 22page

Cancer survivors' experiences of group cognitive behavioural therapy

Section A: Prevalence of anxiety amongst cancer survivors (CS) is high and growing evidence suggests benefits of individual, or group, CBT in reducing anxiety. However, previous reviews were either cancer specific or specified cancer severity. Hence, there is a lack of a review looking at group CBT effectiveness on anxiety across different cancers and severity that this review aims to explore. A systematic review was conducted and twelve RCT studies were reviewed. Results indicated that group CBT interventions were effective in improving anxiety in CS across cancer types. Shortterm interventions also produced positive results. Implications for future research were discussed. Section B: Cancer diagnosis impacts significantly on patients’ anxiety and quality of life. Although studies investigating the effectiveness of group CBT in CS have increased, there remains a paucity of data exploring CS experiences. This study aims to investigate CS’ experiences of receiving group CBT for anxiety. Qualitative grounded theory methodology was applied. Thirteen CS attended a telephone or face-to-face interview. A framework was developed and findings indicated that group CBT seemed acceptable amongst CS, a range of positive and negative experiences were reported and anxiety improved. Some of the mechanisms of change were understanding anxiety, connection with others, accepting cancer, greater hope about the future and access to CBT tools

Majorana Fermions Signatures in Macroscopic Quantum Tunneling

Thermodynamic measurements of magnetic fluxes and I-V characteristics in SQUIDs offer promising paths to the characterization of topological superconducting phases. We consider the problem of macroscopic quantum tunneling in an rf-SQUID in a topological superconducting phase. We show that the topological order shifts the tunneling rates and quantum levels, both in the parity conserving and fluctuating cases. The latter case is argued to actually enhance the signatures in the slowly fluctuating limit, which is expected to take place in the quantum regime of the circuit. In view of recent advances, we also discuss how our results affect a $\pi$-junction loop.Comment: 10 pages, 11 figure