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### Open Access Journals

CPAA

We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and $1/2$-Hölder continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior Hölder continuity estimate.
We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder continuity estimate.
In particular, our results apply to complex perturbations of a single real equation.

DCDS

We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or Hölder continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.

DCDS

By adapting a method in [11] with a suitable modification, we show
that the critical dissipative quasi-geostrophic equations in
$R^2$ has global well-posedness with arbitrary $H^1$
initial data. A decay in time estimate for homogeneous Sobolev norms
of solutions is also discussed.

DCDS

We consider a transport-diffusion equation of the form $\partial_t
\theta + v \cdot \nabla \theta + \nu \mathcal{A} \theta = 0$, where $v$ is a
given time-dependent vector field on $\mathbb R^d$. The operator
$\mathcal{A}$ represents log-modulated fractional dissipation: $\mathcal{A}=\frac
{|\nabla|^{\gamma}}{\log^{\beta}(\lambda+|\nabla|)}$ and the
parameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$,
$\lambda>1$. We introduce a novel nonlocal decomposition of the
operator $\mathcal{A}$ in terms of a weighted integral of the usual
fractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus a
smooth remainder term which corresponds to an $L^1$ kernel. For a
general vector field $v$ (possibly non-divergence-free) we prove a
generalized $L^\infty$ maximum principle of the form $ \|
\theta(t)\|_\infty \le e^{Ct} \| \theta_0 \|_{\infty}$ where the
constant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the same
inequality holds for $\|\theta(t)\|_p$ with $1\le p \le \infty$. Under the additional assumption that $\theta_0\in L^2$, we show that $\|\theta(t)\|_p$ is uniformly bounded for $2\le p\le \infty$. At the cost of a possible exponential factor, this extends a recent result of
Hmidi [7] to the full regime $d\ge 1$, $0\le \gamma \le
2$ and removes the incompressibility assumption in the $L^\infty$
case.

DCDS

We study the critical and super-critical dissipative
quasi-geostrophic equations in $\R^2$ or $\T^2$. An optimal local smoothing effect of solutions with arbitrary initial data in $H^{2-\gamma}$ is proved. As a main application, we establish the
global well-posedness for the critical 2D quasi-geostrophic
equations with periodic $H^1$ data. Some decay in time estimates are
also provided.

DCDS

We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables.
Our results can be applied to Neumann boundary value problems for

*stochastic*partial differential equations with BMO$_x$ coefficients.
DCDS

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincaré inequality on Reifenberg flat domains, the proof of which is of independent interest.

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