Contents
Research Article
On the Cauchy problem associated with the Brinkman flow in
$\mathbb {R}_{+}^{3}$
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- Published online by Cambridge University Press: 07 September 2021, pp. 1089-1108
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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.
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Global regularity for the micropolar Rayleigh-Bénard problem with only velocity dissipation
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- Published online by Cambridge University Press: 26 August 2021, pp. 1109-1138
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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.
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Riemannian approximation in Carnot groups
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- Published online by Cambridge University Press: 06 September 2021, pp. 1139-1154
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Qualitative properties of singular solutions to fractional elliptic equations
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- Published online by Cambridge University Press: 14 September 2021, pp. 1155-1190
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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.
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Partial regularity for minimizers of discontinuous quasiconvex integrals with general growth
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- Published online by Cambridge University Press: 01 December 2021, pp. 1191-1232
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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.
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Asymptotic behaviour of positive solutions of semilinear elliptic problems with increasing powers
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- Published online by Cambridge University Press: 28 September 2021, pp. 1233-1250
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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.
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Well-posedness of the free surface problem on a Newtonian fluid between cylinders rotating at different speeds
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- Published online by Cambridge University Press: 21 September 2021, pp. 1251-1276
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Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor
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- Published online by Cambridge University Press: 21 September 2021, pp. 1277-1290
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Global well-posedness and decay estimates for three-dimensional compressible Navier–Stokes–Allen–Cahn systems
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- Published online by Cambridge University Press: 11 October 2021, pp. 1291-1322
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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.
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The Tutte polynomial and toric Nakajima quiver varieties
- Published online by Cambridge University Press: 27 October 2021, pp. 1323-1339
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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.
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Topological volumes of fibrations: a note on open covers
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- Published online by Cambridge University Press: 11 October 2021, pp. 1340-1360
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Front Cover (OFC, IFC) and matter
PRM volume 152 issue 5 Cover and Front matter
- Published online by Cambridge University Press: 23 September 2022, pp. f1-f2
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Back Cover (IBC, OBC) and matter
PRM volume 152 issue 5 Cover and Back matter
- Published online by Cambridge University Press: 23 September 2022, pp. b1-b2
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