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In this paper we give a simple proof of the existence of global-in-time smooth solutions for the convective Brinkman-Forchheimer equations (also called in the literature the tamed Navier-Stokes equations) $$ \partial_tu -\mu\Delta u + (u \cdot \nabla)u + \nabla p + \alpha u + \beta|u|^{r - 1}u = 0 $$ on a $3$D periodic domain, for values of the absorption exponent $r$ larger than $3$. Furthermore, we prove that global, regular solutions exist also for the critical value of exponent $r = 3$, provided that the coefficients satisfy the relation $4\mu\beta \geq 1$. Additionally, we show that in the critical case every weak solution verifies the energy equality and hence is continuous into the phase space $L^2$. As an application of this result we prove the existence of a strong global attractor, using the theory of evolutionary systems developed by Cheskidov., Comment: 17 pages

The aim of this paper is to prove energy conservation for the incompressible Euler equations in a domain with boundary. We work in the domain $\mathbb{T}^2\times\mathbb{R}_+$, where the boundary is both flat and has finite measure. However, first we study the equations on domains without boundary (the whole space $\mathbb{R}^3$, the torus $\mathbb{T}^3$, and the hybrid space $\mathbb{T}^2\times\mathbb{R}$). We make use of some of the arguments of Duchon \& Robert ({\it Nonlinearity} {\bf 13} (2000) 249--255) to prove energy conservation under the assumption that $u\in L^3(0,T;L^3(\mathbb{R}^3))$ and one of the two integral conditions \begin{equation*} \lim_{|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\mathbb{R}^3} |u(x+y)-u(x)|^3\,d x\,d t=0 \end{equation*} or \begin{equation*} \int_0^T\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{|u(x)-u(y)|^3}{|x-y|^{4+\delta}}\,d x\,d y<\infty,\qquad\delta>0, \end{equation*} the second of which is equivalent to requiring $u\in L^3(0,T;W^{\alpha,3}(\mathbb{R}^3))$ for some $\alpha>1/3$. We then use the first of these two conditions to prove energy conservation for a weak solution $u$ on $D_+:=\mathbb{T}^2\times \mathbb{R}_+$: we extend $u$ a solution defined on the whole of $\mathbb{T}^2\times\mathbb{R}$ and then use the condition on this domain to prove energy conservation for a weak solution $u\in L^3(0,T;L^3(D_+))$ that satisfies \begin{equation*} \lim_{|y|\to 0} \frac{1}{|y|}\int^{T}_{0}\iint_{\mathbb{T}^2}\int^\infty_{|y|}|u(t,x+y)-u(t,x)|^3 \,d x_3 \,d x_1 \,d x_2 \,d t=0, \end{equation*} and certain continuity conditions near the boundary $\partial D_+=\{x_3=0\}$.

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^d$, $d=2,3$, with initial data $B_0\in H^s(\mathbb{R}^d)$ and $u_0\in H^{s-1+\varepsilon}(\mathbb{R}^d)$ for $s>d/2$ and any $0<\varepsilon<1$. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking $\varepsilon=0$ is explained by the failure of solutions of the heat equation with initial data $u_0\in H^{s-1}$ to satisfy $u\in L^1(0,T;H^{s+1})$; we provide an explicit example of this phenomenon.

We study the continuity of pullback and uniform attractors for non-autonomous dynamical systems with respect to perturbations of a parameter. Consider a family of dynamical systems parameterised by a complete metric space $\Lambda$ such that for each $\lambda\in\Lambda$ there exists a unique pullback attractor $\mathcal A_\lambda(t)$. Using the theory of Baire category we show under natural conditions that there exists a residual set $\Lambda_*\subseteq\Lambda$ such that for every $t\in\mathbb R$ the function $\lambda\mapsto\mathcal A_\lambda(t)$ is continuous at each $\lambda\in\Lambda_*$ with respect to the Hausdorff metric. Similarly, given a family of uniform attractors $\mathbb A_\lambda$, there is a residual set at which the map $\lambda\mapsto\mathbb A_\lambda$ is continuous. We also introduce notions of equi-attraction suitable for pullback and uniform attractors and then show when $\Lambda$ is compact that the continuity of pullback attractors and uniform attractors with respect to $\lambda$ is equivalent to pullback equi-attraction and, respectively, uniform equi-attraction. These abstract results are then illustrated in the context of the Lorenz equations and the two-dimensional Navier-Stokes equations.

In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates., Comment: 15 pages, to appear in "Recent Progress in the Theory of the Euler and Navier--Stokes Equations", eds. J.C. Robinson, J.L. Rodrigo, W. Sadowski and A. Vidal-L\'opez, Cambridge University Press, 2016

Generalized differential forms are employed to construct generalized connections. Lorentzian four-metrics determined by certain of these connections satisfy Einstein's vacuum field equations when the connections are flat. Generalized Chern-Simons action principles with Einstein's equations as Euler-Lagrange equations are constructed by using these connections., Comment: 14 pages

If $u$ is a smooth solution of the Navier--Stokes equations on ${\mathbb R}^3$ with first blowup time $T$, we prove lower bounds for $u$ in the Sobolev spaces $\dot H^{3/2}$, $\dot H^{5/2}$, and the Besov space $\dot B^{5/2}_{2,1}$, with optimal rates of blowup: we prove the strong lower bounds $\|u(t)\|_{\dot H^{3/2}}\ge c(T-t)^{-1/2}$ and $\|u(t)\|_{\dot B^{5/2}_{2,1}}\ge c(T-t)^{-1}$, but in $\dot H^{5/2}$ we only obtain the weaker result $\limsup_{t\to T^-}(T-t)\|u(t)\|_{\dot H^{5/2}}\ge c$. The proofs involve new inequalities for the nonlinear term in Sobolev and Besov spaces, both of which are obtained using a dyadic decomposition of $u$.

In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of $\mathbb{R}^{n}$, $n=2,3$, for divergence-free initial data in certain Besov spaces, namely $\boldsymbol{u}_{0} \in B^{n/2-1}_{2,1}$ and $\boldsymbol{B}_{0} \in B^{n/2}_{2,1}$. The a priori estimates include the term $\int_{0}^{t} \| \boldsymbol{u}(s) \|_{H^{n/2}}^{2} \, \mathrm{d} s$ on the right-hand side, which thus requires an auxiliary bound in $H^{n/2-1}$. In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in $H^{1/2}$ is required, which we prove using the splitting method of Calder\'on (Trans. Amer. Math. Soc. 318(1), 179--200, 1990). By contrast, we prove that such solutions are unique in 3D, but the proof of uniqueness in 2D is more difficult and remains open.

We show that self-similar sets arising from iterated function systems that satisfy the Moran open-set condition, a canonical class of fractal sets, are `equi-homogeneous'. This is a regularity property that, roughly speaking, means that at each fixed length-scale any two neighbourhoods of the set have covers of approximately equal cardinality. Self-similar sets are notable in that they are Ahlfors-David regular, which implies that their Assouad and box-counting dimensions coincide. More generally, attractors of non-autonomous iterated functions systems (where maps are allowed to vary between iterations) can have distinct Assouad and box-counting dimensions. Consequently the familiar notion of Ahlfors-David regularity is too strong to be useful in the analysis of this important class of sets, which include generalised Cantor sets and possess different dimensional behaviour at different length-scales. We further develop the theory of equi-homogeneity showing that it is a weaker property than Ahlfors-David regularity and distinct from any previously defined notion of dimensional equivalence. However, we show that if the upper and lower box-counting dimensions of an equi-homogeneous set are equal and `attained' in a sense we make precise then the lower Assouad, Hausdorff, packing, lower box-counting, upper box-counting and Assouad dimensions coincide. Our main results provide conditions under which the attractor of a non-autonomous iterated function system is equi-homogeneous and we use this to compute the Assouad dimension of a certain class of these highly non-trivial sets.

In this paper we treat three problems on a two-dimensional `punctured periodic domain': we take $\Omega_r=(-L,L)^2\setminus D_r$, where $D_r=B(0,r)$ is the disc of radius $r$ centred at the origin. We impose periodic boundary conditions on the boundary of the box $\Omega=(-L,L)^2$, and Dirichlet boundary conditions on the circumference of the disc. In this setting we consider the Poisson equation, the Stokes equations, and the time-dependent Navier-Stokes equations, all with a fixed forcing function $f$ (which must satisfy $\int_\Omega f=0$ for the stationary problems), and examine the behaviour of solutions as $r\to0$. In all three cases we show convergence of the solutions to those of the limiting problem, i.e.\ the problem posed on all of $\Omega$ with periodic boundary conditions.

A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least $(d+1)/2$ (better estimates are available in the literature). We also study the `generic validity' of the Kakeya conjecture in the setting of Baire Category and prove that typical Besicovitch sets have full upper box-counting dimension. We also study a weaker version of the Kakeya problem where unit line segments are replaced by half-infinite lines. We prove that such `half-extended Besicovitch sets' have full Assouad dimension. This can be viewed as full resolution of a (much weakened) version of the Kakeya problem., Comment: 17 pages. Incorporated referee suggestions, improved results in section 4, to appear in Real Analysis Exchange

Let $\Lambda$ be a complete metric space, and let $\{S_\lambda(\cdot):\ \lambda\in\Lambda\}$ be a parametrised family of semigroups with global attractors ${\mathscr A}_\lambda$. We assume that there exists a fixed bounded set $D$ such that ${\mathscr A}_\lambda\subset D$ for every $\lambda\in\Lambda$. By viewing the attractors as the limit as $t\to\infty$ of the sets $S_\lambda(t)D$, we give simple proofs of the equivalence of `equi-attraction' to continuity (when this convergence is uniform in $\lambda$) and show that the attractors ${\mathscr A}_\lambda$ are continuous in $\lambda$ at a residual set of parameters in the sense of Baire Category (when the convergence is only pointwise).

We consider the scalar semilinear heat equation $u_t-\Delta u=f(u)$, where $f\colon[0,\infty)\to[0,\infty)$ is continuous and non-decreasing but need not be convex. We completely characterise those functions $f$ for which the equation has a local solution bounded in $L^q(\Omega)$ for all non-negative initial data $u_0\in L^q(\Omega)$, when $\Omega\subset{\mathbb R}^d$ is a bounded domain with Dirichlet boundary conditions. For $q\in(1,\infty)$ this holds if and only if $\limsup_{s\to\infty}s^{-(1+2q/d)}f(s)<\infty$; and for $q=1$ if and only if $\int_1^\infty s^{-(1+2/d)}F(s)\,{\rm d}s<\infty$, where $F(s)=\sup_{1\le t\le s}f(t)/t$. This shows for the first time that the model nonlinearity $f(u)=u^{1+2q/d}$ is truly the `boundary case' when $q\in(1,\infty)$, but that this is not true for $q=1$. The same characterisation results hold for the equation posed on the whole space ${\mathbb R}^d$ provided that in addition $\limsup_{s\to0}f(s)/s<\infty$.

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of `equi-homogeneity' of a set, which requires a uniformity in the size of local covers at all lengths and at all points. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set `attains' these dimensions (analogous to `s-sets' when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any $\alpha\in(0,1)$ and any $\beta,\gamma\in(0,1)$ such that $\beta + \gamma\geq 1$ we can construct two generalised Cantor sets $C$ and $D$ such that $\text{dim}_{B}C=\alpha\beta$, $\text{dim}_{B}D=\alpha\gamma$, and $\text{dim}_{A}C=\text{dim}_{A}D=\text{dim}_{A}(C\times D)=\text{dim}_{B}(C\times D)=\alpha$.

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the \emph{weak separation property} is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the \emph{weak separation property} is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the \emph{weak separation property} is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a \emph{weak tangent} that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the `equality/maximal' dichotomy does not extend to this setting., Comment: 24 pages, 2 figures

In 2000 Constantin showed that the incompressible Euler equations can be written in an "Eulerian-Lagrangian" form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain H\"older spaces $C^{1,\mu}$. We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data in $H^s$ for $n\geq2$ and $s>\frac{n}{2}+1$, a unique local-in-time solution exists on the $n$-torus that is continuous into $H^s$ and $C^1$ into $H^{s-1}$. These solutions automatically have $C^1$ trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian-Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory., Comment: 17 pages, to appear in J. Math. Fluid Mech. Lemmas 4 and 6 revised, several minor changes

This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^{n}$, $n=2, 3$, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891-907, 1988).