Higher Sobolev Regularity of Convex Integration Solutions in Elasticity: The Dirichlet Problem with Affine Data in int(Klc)\text{int}(K^{lc})

In this article we continue our study of higher Sobolev regularity of flexible convex integration solutions to differential inclusions arising from applications in materials sciences. We present a general framework yielding higher Sobolev regularity for Dirichlet problems with affine data in $\text{int}(K^{lc})$. This allows us to simultaneously deal with linear and nonlinear differential inclusion problems. We show that the derived higher integrability and differentiability exponent has a lower bound, which is independent of the position of the Dirichlet boundary data in $\text{int}(K^{lc})$. As applications we discuss the regularity of weak isometric immersions in two and three dimensions as well as the differential inclusion problem for the geometrically linear hexagonal-to-rhombic and the cubic-to-orthorhombic phase transformations occurring in shape memory alloys.Comment: 50 pages, 13 figure

On the Boussinesq equations with non-monotone temperature profiles

In this article we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles $T(y)$. As a first main result we show that if $T'$ is of size at most $\nu^{1/3}$ in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh-B\'enard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.Comment: 30 pages; updated and added reference

On stability estimates for the inviscid Boussinesq equations

We consider the (in)stability problem of the inviscid 2D Boussinesq equations near a combination of a shear flow $v = (y, 0)$ and a stratified temperature $\theta=\alpha y$ with $\alpha>\frac{1}{4}$. We show that for any $\epsilon>0$ there exist non-trivial explicit solutions, which are initially perturbations of size $\epsilon$, and grow to size $1$ on a time scale $\epsilon^{-2}$. Moreover, the (simplified) linearized problem around these non-trivial states exhibits improved upper bounds on the possible size of norm inflation for frequencies larger and smaller than $\epsilon^{−4}$

Hamid’s Travelogue. Mimetic Transformations and Spiritual Connectivities Across Mediterranean Topographies of Grace

In their seminal work that helped to re-invent Mediterranean anthropology some 20 years ago, Horden and Purcell argue that the religious landscape reflects both, the fragmented topography of Mediterranean micro-regions and the means by which the fragmentation is overcome. In order to explore how space and time concern the divine along and across Mediterranean shores, this paper examines how social and spiritual borders are crossed in religious practice and how graduated socialities are generated, shaped and negotiated. It argues that connectivities, lateral and vertical, are forged or undone by turning borders into thresholds and vice-versa. Drawing from both, the history of Mediterranean anthropology of religion and ethnographic material from transnational mobile members of trance networks, the paper sketches an anthropology of blessing across nested fields of exteriority and alterity, found within and without the social niches of Mediterranean lifeworlds

On the Boussinesq equations with non-monotone temperature profiles

Abstract. In this article we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles $T(y)$. As a first main result we show that if $T′$ is of size at most $\nu^{1/3}$ in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh-Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity

On echo chains in the linearized Boussinesq equations around traveling waves

We consider the 2D Boussinesq equations with viscous but without thermal dissipation and observe that in any neighborhood of Couette flow and hydrostatic balance (with respect to local norms) there are time-dependent traveling wave solutions of the form $\omega=-1+f(t)\cos(x-ty)$, $\theta=\alpha y+g(t)\sin(x-ty)$. As our main result we show that the linearized equations around these waves for $\alpha=0$ exhibit echo chains and norm inflation despite viscous dissipation of the velocity. Furthermore, we construct initial data in a critical Gevrey 3 class, for which temperature and vorticity diverge to infinity in Sobolev regularity as $t to infty$ but for which the velocity still converges

On the forced Euler and Navier-Stokes equations: Linear damping and modified scattering

We study the asymptotic behavior of the forced linear Euler and nonlinear Navier-Stokes equations close to Couette flow in a periodic channel. As our main result we show that for smooth time-periodic forcing linear inviscid damping persists, i.e. the velocity field (weakly) asymptotically converges. However, stability and scattering to the transport problem fail in $H^{s}, s>-1$. We further show that this behavior is consistent with the nonlinear Euler equations and that a similar result also holds for the nonlinear Navier-Stokes equations. Hence, these results provide an indication that nonlinear inviscid damping may still hold in Sobolev regularity in the above sense despite the Gevrey regularity instability results of [Deng-Masmoudi 2018].Comment: 26 page