Global well-posedness for a slightly supercritical surface quasi-geostrophic equation

We use a nonlocal maximum principle to prove the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation. By this we mean that the velocity field $u$ is obtained from the active scalar $\theta$ by a Fourier multiplier with symbol $i k^\perp |k|^{-1} m(k|)$, where $m$ is a smooth increasing function that grows slower than $\log \log |k|$ as $|k|\rightarrow \infty$.Comment: 11 pages, second version with slightly stronger resul

Holder continuity for a drift-diffusion equation with pressure

We address the persistence of H\"older continuity for weak solutions of the linear drift-diffusion equation with nonlocal pressure u_t + b \cdot \grad u - \lap u = \grad p,\qquad \grad\cdot u =0 on $[0,\infty) \times \R^{n}$, with $n \geq 2$. The drift velocity $b$ is assumed to be at the critical regularity level, with respect to the natural scaling of the equations. The proof draws on Campanato's characterization of H\"older spaces, and uses a maximum-principle-type argument by which we control the growth in time of certain local averages of $u$. We provide an estimate that does not depend on any local smallness condition on the vector field $b$, but only on scale invariant quantities

Global well-posedness for an advection-diffusion equation arising in magneto-geostrophic dynamics

We use De Giorgi techniques to prove H\"older continuity of weak solutions to a class of drift-diffusion equations, with $L^2$ initial data and divergence free drift velocity that lies in $L_{t}^{\infty}BMO_{x}^{-1}$. We apply this result to prove global regularity for a family of active scalar equations which includes the advection-diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earth's fluid core.Comment: To appear in Annales de l'Institut Henri Poincare - Analyse non lineair

On the loss of continuity for super-critical drift-diffusion equations

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts, that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local $L^1$ norm of the drift, which is a super-critical quantity.Comment: Minor edit

Nonlinear Instability for the Critically Dissipative Quasi-Geostrophic Equation

We prove that linear instability implies non-linear instability in the energy norm for the critically dissipative quasi-geostrophic equation.Comment: 16 pages, corrected typos, a global bound that was obtained for the unforced equation by Kiselev-Nazarov-Volberg obtained for the forced equation and utilized in the paper

Gate-tunable high mobility remote-doped InSb/In_1-xAl_xSb quantum well heterostructures

Gate-tunable high-mobility InSb/In_{1-x}Al_{x}Sb quantum wells (QWs) grown on GaAs substrates are reported. The QW two-dimensional electron gas (2DEG) channel mobility in excess of 200,000 cm^{2}/Vs is measured at T=1.8K. In asymmetrically remote-doped samples with an HfO_{2} gate dielectric formed by atomic layer deposition, parallel conduction is eliminated and complete 2DEG channel depletion is reached with minimal hysteresis in gate bias response of the 2DEG electron density. The integer quantum Hall effect with Landau level filling factor down to 1 is observed. A high-transparency non-alloyed Ohmic contact to the 2DEG with contact resistance below 1{\Omega} \cdot mm is achieved at 1.8K.Comment: 25 pages, 10 figure