Endogenous Exchange Rate Regime Switches

In this paper we demonstrate that exchange rate regime switching is compatible with optimal government policies. Nominal exchange-rate regimes are formalized as equilibrium commitments on future seigniorage policies, and the collapse of an exchange-rate peg as an excusable default which allows the government to lump-sum tax private sector money holdings. We demonstrate that a regime in which the exchange-rate peg is allowed to collapse when government spending is unusually high is a trigger-strategy equilibrium. Such a regime can be superior to both fixed and flexible exchange rate because it combines some of the flexibility of the floating exchange rates with some of the benefits of precommitment afforded by fixed rates.

A profile decomposition approach to the L∞/t (L3/ x) Navier–Stokes regularity criterion

In this paper we continue to develop an alternative viewpoint on recent studies of Navier–Stokes regularity in critical spaces, a program which was started in the recent work by Kenig and Koch (Ann Inst H Poincaré Anal Non Linéaire 28(2):159–187, 2011). Specifically, we prove that strong solutions which remain bounded in the space L3(R3) do not become singular in finite time, a known result established by Escauriaza et al. (Uspekhi Mat Nauk 58(2(350)):3–44, 2003) in the context of suitable weak solutions. Here, we use the method of “critical elements” which was recently developed by Kenig and Merle to treat critical dispersive equations. Our main tool is a “profile decomposition” for the Navier–Stokes equations in critical Besov spaces which we develop here. As a byproduct of this tool, assuming a singularity-producing initial datum for Navier–Stokes exists in a critical Lebesgue or Besov space, we show there is one with minimal norm, generalizing a result of Rusin and Sverak (J Funct Anal 260(3):879–891, 2011)

Parabolic fractal dimension of forward-singularities for Navier-Stokes and liquid crystals inequalities

In 1985, V. Scheffer discussed partial regularity results for what he called solutions to the "Navier-Stokes inequality". These maps essentially satisfy the incompressibility condition as well as the local and global energy inequalities and the pressure equation which may be derived formally from the Navier-Stokes system of equations, but they are not required to satisfy the Navier-Stokes system itself. One may extend this notion to a system considered by Fang-Hua Lin and Chun Liu in the mid 1990s related to models of the flow of nematic liquid crystals, which include the Navier-Stokes system when the "director field" $d$ is taken to be zero. In addition to an extended Navier-Stokes system, the Lin-Liu model includes a further parabolic system which implies an a priori maximum principle for $d$, which is lost when one considers the analogous 'inequality'. In 2018, Qiao Liu proved a partial regularity result for certain solutions to the Lin-Liu model in terms of the parabolic fractal dimension $\textrm{dim}_{\textrm{pf}}$, relying on the boundedness of $d$ coming from the maximum principle. Specifically, Q. Liu proves $\textrm{dim}_{\textrm{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}$ for any compact $\mathcal{K}$, where $\Sigma_{-}$ is the set of "forward-singular" space-time points, near which the solution blows up forwards in time. For solutions to the corresponding "inequality", we prove that, without any compensation for the lack of maximum principle, one has $\textrm{dim}_{\textrm{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac {55}{13}$. We also provide a range of criteria, including as just one example the boundedness of $d$, any one of which would furthermore imply that $\textrm{dim}_{\textrm{pf}}(\Sigma_{-} \cap \mathcal{K}) \leq \tfrac{95}{63}$ for solutions to the inequality, just as Q. Liu proved for solutions to the Lin-Liu system itself.Comment: 20 pages. arXiv admin note: text overlap with arXiv:2001.0409

A general wavelet-based profile decomposition in critical embedding of function spaces

International audienceWe characterize the lack of compactness in the critical embedding of functions spaces X Y having similar scaling properties in the following terms : a sequence (un)n 0 bounded in X has a subsequence that can be expressed as a finite sum of translations and dilations of functions ( l)l>0 such that the remainder converges to zero in Y as the number of functions in the sum and n tend to +1. Such a decomposition was established by G'erard in [13] for the embedding of the homogeneous Sobolev space X = ˙H s into the Y = Lp in d dimensions with 0 < s = d/2 − d/p, and then generalized by Jaffard in [15] to the case where X is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces X and Y that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of X and Y satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H¨older and BMO spaces

Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by ϵ∈[1,2]\boldsymbol{\epsilon \in [1,2]}

For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+\epsilon}(\mathbb{R}^n)}\gtrsim_{\varphi,\epsilon,(p\vee q\vee 2)}{(T-t)}^{-\epsilon/2}$ for all $\epsilon\in[1,2)$ and $p,q\in[1,\frac{n}{2-\epsilon})$, where $s_{p}:=-1+\frac{n}{p}$ is the scaling-critical regularity, and $\varphi$ is the cutoff function used to define the Littlewood-Paley projections. For $\epsilon =2$, we prove the same type of estimate but only for $q=1$: ${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^n)}\gtrsim_{\varphi,(p\vee 2)}{(T-t)}^{-1}$ for all $p\in [1,\infty)$. Under the additional restriction that $p,q\in[1,2]$ and $n=3$, these blowup estimates are implied by those first proved by Robinson, Sadowski and Silva (J. Math. Phys., 2012) for $p=q=2$ in the case $\epsilon\in(1,2)$, and by McCormick, Olson, Robinson, Rodrigo, Vidal-L\'{o}pez and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases $(\epsilon,q)=(1,2)$ and $(\epsilon,q)=(2,1)$.Comment: 10 page

Local well-posedness and blowup rates for the Navier-Stokes equations in subcritical Lorentz spaces

We develop a local-in-time theory for mild solutions to the Navier-Stokes equations in subcritical Lorentz spaces, establishing uniqueness, local existence and continuity of solutions. We show that if a solution $u$ has a finite blowup time $T$, then the Lorentz quasinorm ${\|u(t)\|}_{L^{p,q}(\mathbb{R}^{n})}^{*}$ satisfies ${\|u(t)\|}_{L^{p,q}(\mathbb{R}^{n})}^{*} \gtrsim_{n,p} {(T-t)}^{-\frac{1}{2}\left(1-\frac{n}{p}\right)}$ for all $p\in(n,\infty]$, $q\in [1,\infty]$ and $t\in(0,T)$, and we obtain explicit bounds on the implied constant. This extends a well-known result established by J. Leray (1934) in the Lebesgue setting $q=p$.Comment: 20 pages; list of references updated and expanded in this versio

The Homeostasis Protocol: Avoiding Transaction Coordination Through Program Analysis

Datastores today rely on distribution and replication to achieve improved performance and fault-tolerance. But correctness of many applications depends on strong consistency properties - something that can impose substantial overheads, since it requires coordinating the behavior of multiple nodes. This paper describes a new approach to achieving strong consistency in distributed systems while minimizing communication between nodes. The key insight is to allow the state of the system to be inconsistent during execution, as long as this inconsistency is bounded and does not affect transaction correctness. In contrast to previous work, our approach uses program analysis to extract semantic information about permissible levels of inconsistency and is fully automated. We then employ a novel homeostasis protocol to allow sites to operate independently, without communicating, as long as any inconsistency is governed by appropriate treaties between the nodes. We discuss mechanisms for optimizing treaties based on workload characteristics to minimize communication, as well as a prototype implementation and experiments that demonstrate the benefits of our approach on common transactional benchmarks

A field study on the influence of food and immune priming on a bumblebee-gut parasite system

Laboratory experiments are often preferred over field experiments because they allow the control of confounding factors that would otherwise influence the causal effect of a particular focal experimental factor. These confounding factors can, however, significantly alter the response of an organism confronted with a particular situation, which can have great implications. In a field experiment with a bumblebee host-parasite system, we looked at the influence of additional food supply and immune challenge on various colony fitness values and parasite traits. We could confirm the importance of food on the colony fitness, but not on parasite infection probability or parasite genetic diversity. In contrast to the findings of laboratory experiments of this system, challenge of the immune system had no significant effect on colony fitness or parasite infections. These results likely reflect an overriding effect of environmental variation without disproving the concept of a cost of defence per se. But the results also demonstrate that confounding factors purposely controlled for in the laboratory have to be weighed against their ecological relevance, and stress the need for careful analysis before any direct transfer is made of laboratory results to field situation