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$\mathbb {R}_{+}^{3}$
In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space
under the assumption that the plane 
\[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \]
$z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.
This paper is concerned with the global regularity problem on the micropolar Rayleigh-Bénard problem with only velocity dissipation in 
$\mathbb {R}^{d}$ with 
$d=2\ or\ 3$. By fully exploiting the special structure of the system, introducing two combined quantities and using the technique of Littlewood-Paley decomposition, we establish the global regularity of solutions to this system in 
$\mathbb {R}^{2}$. Moreover, we obtain the global regularity for fractional hyperviscosity case in 
$\mathbb {R}^{3}$ by employing various techniques including energy methods, the regularization of generalized heat operators on the Fourier frequency localized functions and logarithmic Sobolev interpolation inequalities.
In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations
where 
\begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*}
$0<\alpha <1$, 
$\Omega = \mathbb {R}^{N}$ or 
$\Omega$ is a smooth bounded domain, 
$\Gamma$ is a singular subset of 
$\Omega$ with fractional capacity zero, 
$f(t)$ is locally bounded and positive for 
$t\in [0,\,\infty )$, and 
$f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in 
$t$ for large 
$t$, rather than for every 
$t>0$. Our main result is that the solutions satisfy the estimate
This estimate is new even for 
\begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*}
$\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.
We prove the partial Hölder continuity for minimizers of quasiconvex functionals
where 
\begin{equation*} \mathcal{F}({\bf u}) \colon =\int_{\Omega} f(x,{\bf u},D{\bf u})\,\textrm{d} x, \end{equation*}
$f$ satisfies a uniform VMO condition with respect to the 
$x$-variable and is continuous with respect to 
${\bf u}$. The growth condition with respect to the gradient variable is assumed a general one.
We prove existence results of two solutions of the problem
where 
\[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \]
$L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, 
$p\in (2,2^{*}]$ and 
$\lambda$ and 
$m$ sufficiently large. Then their asymptotical limit as 
$m\to +\infty$ is investigated showing different behaviours.
We study the small data global well-posedness and time-decay rates of solutions to the Cauchy problem for three-dimensional compressible Navier–Stokes–Allen–Cahn equations via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, the 
$\dot {H}^{-s}$(
$0\leq s<\frac {3}{2}$) negative Sobolev norms is shown to be preserved along time evolution and enhance the decay rates.
For a quiver 
$Q$ with underlying graph 
$\Gamma$, we take 
$ {\mathcal {M}}$ an associated toric Nakajima quiver variety. In this article, we give a direct relation between a specialization of the Tutte polynomial of 
$\Gamma$, the Kac polynomial of 
$Q$ and the Poincaré polynomial of 
$ {\mathcal {M}}$. We do this by giving a cell decomposition of 
$ {\mathcal {M}}$ indexed by spanning trees of 
$\Gamma$ and ‘geometrizing’ the deletion and contraction operators on graphs. These relations have been previously established in Hausel–Sturmfels [6] and Crawley-Boevey–Van den Bergh [3], however the methods here are more hands-on.