Critical Norm Research Articles

On the Critical Norm Concentration for the Inhomogeneous Nonlinear Schrödinger Equation

We consider the inhomogeneous nonlinear Schrödinger equation (INLS) in \({\mathbb {R}}^N\)$$\begin{aligned} i \partial _t u + \Delta u + |x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$and show the \(L^2\)-norm concentration for the finite time blow-up solutions in the \(L^2\)-critical case, \(\sigma =\frac{2-b}{N}\). Moreover, we provide an alternative proof for the classification of minimal mass blow-up solutions, first proved by Genoud and Combet (J Evol Equ 16(2):483–500, 2016, https://doi.org/10.1007/s00028-015-0309-z). For the case \(\frac{2-b}{N}< \sigma < \frac{2-b}{N-2}\), we show results regarding the \(L^p\)-critical norm concentration, generalizing the argument of Holmer and Roudenko (Appl Math Res eXpress 2007(1):Art. ID abm004, 2007) to the INLS setting.

Reducing the O(3) model as an effective field theory

We consider the O(3) or ℂP1 nonlinear sigma model as an effective field theory in a derivative expansion, with the most general Lagrangian that obeys O(3), parity and Lorentz symmetry. We work out the complete list of possible operators (terms) in the Lagrangian and eliminate as many as possible using integrations by parts. We further show at the four-derivative level, that the theory can be shown to avoid the Ostrogradsky instability, because the dependence on the d’Alembertian operator or so-called box, can be eliminated by a field redefinition. Going to the six-derivative order in the derivative expansion, we show that this can no longer be done, unless we are willing to sacrifice Lorentz invariance. By doing so, we can eliminate all dependence on double time derivatives and hence the Ostrogradsky instability or ghost, however, we unveil a remaining dynamical instability that takes the form either as a spiral instability or a runaway instability and estimate the critical field norm, at which the instability sets off.

Indonesia's Promotion of UN Migrant Protection Norms in ASEAN

Scholars have devoted insufficient attention to Indonesia's foreign policy on migrant worker protection, especially as mobilized in multilateral institutions. This article addresses such knowledge gaps by analyzing why Indonesia has, for almost two decades, persistently promoted the United Nations Migrant Worker Convention in the Association of Southeast Asian Nations (ASEAN) despite constant pushback from migrant-receiving countries. It argues that Indonesia's persistence is driven by its locally constituted meaning of migrant worker rights. In particular, this article advances the critical norms approach in international relations to demonstrate that its interpretation is influenced by "Indonesia's normative baggage," or past experiences with labour migration that have too frequently dealt with the exploitation of Indonesian citizens abroad. This normative baggage in turn shapes the country's diplomacy and promotion of convention standards deemed appropriate for safeguarding Indonesian migrants in ASEAN. In presenting the argument, this article contributes to the study of labour migration by scrutinizing Indonesia's foreign policy on migrant protection and unpacking norm interpretation processes that are necessary in international negotiations.

Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

We consider the inhomogeneous biharmonic nonlinear Schr dinger (IBNLS) equation in $${\mathbb {R}}^N$$ , $$\begin{aligned} i \partial _t u +\Delta ^2 u -|x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$ where $$\sigma > 0$$ and $$b > 0$$ . We first study the local well-posedness in $${\dot{H}}^{s_c}\cap \dot{H}^2 $$ , for $$N\ge 5$$ and $$0<s_c<2$$ , where $$s_c=\frac{N}{2}-\frac{4-b}{2\sigma }$$ . Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $$\dot{H}^{s_c}\cap \dot{H}^2$$ with $$0\le s_c<2$$ . Finally, we study the phenomenon of $$L^{\sigma _c}$$ -norm concentration for finite time blow up solutions with bounded $$\dot{H}^{s_c}$$ -norm, where $$\sigma _c=\frac{2N\sigma }{4-b}$$ . Our main tool is the compact embedding of $$\dot{L}^p\cap \dot{H}^2$$ into a weighted $$L^{2\sigma +2}$$ space, which may be seen of independent interest.

Improved Quantitative Regularity for the Navier–Stokes Equations in a Scale of Critical Spaces

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty L_x^q}<\infty$ where $r=\sqrt{x_1^2+x_2^2}$ and either $q\in(3,\infty)$, or $u$ is axisymmetric and $q\in(2,3]$. Using the strategy of Tao (2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We make use of some tools such as a decomposition of the solution that allows us to use energy methods in these spaces, as well as a Carleman inequality for the heat equation suited for proving quantitative backward uniqueness in cylindrical regions.

Replacing Bureaucrats with Automated Sorcerers?

Abstract Increasingly, federal agencies employ artificial intelligence to help direct their enforcement efforts, adjudicate claims and other matters, and craft regulations or regulatory approaches. Theoretically, artificial intelligence could enable agencies to address endemic problems, most notably 1) the inconsistent decision-making and departure from policy attributable to low-level officials' exercise of discretion; and 2) the imprecise nature of agency rules. But two characteristics of artificial intelligence, its opaqueness and the nonintuitive nature of its correlations, threaten core values of administrative law. Administrative law reflects the principles that 1) persons be judged individually according to announced criteria; 2) administrative regulations reflect some means-end rationality; and 3) administrative decisions be subject to review by external actors and transparent to the public. Artificial intelligence has adverse implications for all three of those critical norms. The resultant tension, at least for now, will constrain administrative agencies' most ambitious potential uses of artificial intelligence.

Quantitative Regularity for the Navier\u2013Stokes Equations Via Spatial Concentration

This paper is concerned with quantitative estimates for the Navier–Stokes equations. First we investigate the relation of quantitative bounds to the behavior of critical norms near a potential singularity with Type I bound Vert uVert _{L^{infty }_{t}L^{3,infty }_{x}}le M. Namely, we show that if T^* is a first blow-up time and (0,T^*) is a singular point then ‖u(·,t)‖L3(B0(R))≥C(M)log(1T∗-t),R=O((T∗-t)12-).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\Vert u(\\cdot ,t)\\Vert _{L^{3}(B_{0}(R))}\\ge C(M)\\log \\Big (\\frac{1}{T^*-t}\\Big ),\\,\\,\\,\\,\\,\\,R=O((T^*-t)^{\\frac{1}{2}-}). \\end{aligned}$$\\end{document}We demonstrate that this potential blow-up rate is optimal for a certain class of potential non-zero backward discretely self-similar solutions. Second, we quantify the result of Seregin (Commun Math Phys 312(3):833–845, 2012), which says that if u is a smooth finite-energy solution to the Navier–Stokes equations on {mathbb {R}}^3times (0,1) with supn‖u(·,t(n))‖L3(R3)<∞andt(n)↑1,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sup _{n}\\Vert u(\\cdot ,t_{(n)})\\Vert _{L^{3}({\\mathbb {R}}^3)}<\\infty \\,\\,\\,\\text {and}\\,\\,\\,t_{(n)}\\uparrow 1, \\end{aligned}$$\\end{document}then u does not blow-up at t=1. To prove our results we develop a new strategy for proving quantitative bounds for the Navier–Stokes equations. This hinges on local-in-space smoothing results (near the initial time) established by Jia and Šverák (2014), together with quantitative arguments using Carleman inequalities given by Tao (2019). Moreover, the technology developed here enables us in particular to give a quantitative bound for the number of singular points in a Type I blow-up scenario.

Elite rhetoric can undermine democratic norms

Democratic stability depends on citizens on the losing side accepting election outcomes. Can rhetoric by political leaders undermine this norm? Using a panel survey experiment, we evaluate the effects of exposure to multiple statements from former president Donald Trump attacking the legitimacy of the 2020 US presidential election. Although exposure to these statements does not measurably affect general support for political violence or belief in democracy, it erodes trust and confidence in elections and increases belief that the election is rigged among people who approve of Trump's job performance. These results suggest that rhetoric from political elites can undermine respect for critical democratic norms among their supporters.

The Keller\u2013Segel system on bounded convex domains in critical spaces

Consider the classical Keller–Segel system on a bounded convex domain varOmega subset {mathbb {R}}^3. In contrast to previous works it is not assumed that the boundary of varOmega is smooth. It is shown that this system admits a local, strong solution for initial data in critical spaces which extends to a global one provided the data are small enough in this critical norm. Furthermore, it is shown that this system admits for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution.

Optimal non-homogeneous improvements for the series expansion of Hardy’s inequality

We consider the series expansion of the [Formula: see text]-Hardy inequality of [G. Barbatis, S. Filippas and A. Tertikas, Series expansion for [Formula: see text] Hardy inequalities, Indiana Univ. Math. J. 52 (2003) 171–190], in the particular case where the distance is taken from an interior point of a bounded domain in [Formula: see text] and [Formula: see text]. For [Formula: see text] we improve it by adding as a remainder term an optimally weighted critical Sobolev norm, generalizing the [Formula: see text] = 2 result of [S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (2002) 186–233] and settling the open question raised in [G. Barbatis, S. Filippas and A. Tertikas, A unified approach to improved [Formula: see text] Hardy inequalities with best constants, Trans. Amer. Math. Soc. 356 (2004) 2169–2196]. For [Formula: see text] we improve it by adding as a remainder term the optimally weighted Hölder seminorm, extending the Hardy–Morrey inequality of [G. Psaradakis, An optimal Hardy–Morrey inequality, Calc. Var. Partial Differential Equations 45 (2012) 421–441] to the series case.

Influence of silicon, boron and rare-earth metals on corrosion resistance of austenitic chromium-nickel steel

The effect of silicon (in range 0.14 – 0.78 wt. %), boron, and rare-earth metals (REM) on the corrosion resistance of low-carbon austenitic chromium-nickel steel of 03Kh18N11 (AISI 304L) grade was studied. It is shown that all steels in quenched state when tested in boiling 56 and 65 % HNO3 solutions have comparable corrosion rates, which do not exceed the critical norm (0.5 mm/year) in accordance with GOST 6032 – 2017 (State Standard). Testing samples in boiling 27 % HNO3 + 4 g/l Cr+6 solution are susceptible to intergranular corrosion (IGC). The corrosion rate and the penetration depth of IGC increase with increasing silicon concentration from 0.14 to 0.78 wt. %. Study of the effect of nitric acid concentration and test temperature has shown that steel with 0.78 wt. % Si has significant corrosion losses exceeding the critical ones when testing in 56 and 65 % HNO3 solutions with temperature of 120 and 130 °С. But steel with high silicon content (0.78 wt. %) and low carbon concentration (0.020 – 0.022 %) after quenching in a range of 1080 – 1150 °C and tempering at 650 °C does not exceed the critical norm on average corrosion rate. Only 0.01 wt. % increase in carbon concentration leads to a significant (more than 30 times) increase in corrosion rate of sensitized steel. It is shown that microalloying with REM does not impair corrosion resistance of sensitized steel. In contrast to REM, alloying chromium-nickel steel with even a small addition of boron (0.0015 %) reduces steel corrosion resistance by an order of magnitude. Corrosion rate inverse dependence on quenching temperature is observed when, with increasing temperature, corrosion rate of 02Kh18N11GS0.38R steel increases. The effect of silicon (in range 0.14 – 0.78 wt. %), boron, and rare-earth metals (REM) on the corrosion resistance of low-carbon austenitic chromium-nickel steel of 03Kh18N11 (AISI 304L) grade was studied. It is shown that all steels in quenched state when tested in boiling 56 and 65 % HNO3 solutions have comparable corrosion rates, which do not exceed the critical norm (0.5 mm/year) in accordance with GOST 6032 – 2017 (State Standard). Testing samples in boiling 27 % HNO3 + 4 g/l Cr+6 solution are susceptible to intergranular corrosion (IGC). The corrosion rate and the penetration depth of IGC increase with increasing silicon concentration from 0.14 to 0.78 wt. %. Study of the effect of nitric acid concentration and test temperature has shown that steel with 0.78 wt. % Si has significant corrosion losses exceeding the critical ones when testing in 56 and 65 % HNO3 solu tions with temperature of 120 and 130 °С. But steel with high silicon content (0.78 wt. %) and low carbon concentration (0.020 – 0.022 %) after quenching in a range of 1080 – 1150 °C and tempering at 650 °C does not exceed the critical norm on average corrosion rate. Only 0.01 wt. % increase in carbon concentration leads to a significant (more than 30 times) increase in corrosion rate of sensitized steel. It is shown that microalloying with REM does not impair corrosion resistance of sensitized steel. In contrast to REM, alloying chromium-nickel steel with even a small addition of boron (0.0015 %) reduces steel corrosion resistance by an order of magnitude. Corrosion rate inverse dependence on quenching temperature is observed when, with increasing temperature, corrosion rate of 02Kh18N11GS0.38R steel increases.

Constructing and confronting bullying and normality at Finnish comprehensive schools from the perspective of critical bullying studies

ABSTRACT In the field of critical bullying studies, norms and normality have been used to explain bullying in a manner that challenges the mainstream conceptualising of bullying. This study concentrates in a more detailed way on how normality and not-normality are connected to bullying and how the students define not-normality as a reason for bullying. I used ethnographic interviews of 48 students to examine the connection between bullying and normality. I found that the students that were defined as ‘not-normal’ were excluded from the school community, but some of them found ways to confront and negotiate their position in the processes of bullying. By describing students who were bullied as not-normal, the bullying was a means to maintain the norms and normality at the school. Thus, in this article, I name a new field of research, the field of critical bullying studies in which the bullying is connected to the norms and normality and it is thus also a structural issue and not only a question of the behaviour of an individual.

Diffusionism and Beyond in IR Norm Research

ABSTRACT This article suggests the relevance of diffusionism in discussing the past, the present, and the future of the International Relations (IR) norm diffusion literature. The paper argues, thus, that IR norm research has reproduced the diffusionist and beyond-diffusionist mechanisms and epistemologies that the anthropological research on culture diffusion has developed, and this has been consequential for the discipline. Accordingly, the diffusionism dominant in the mainstream IR norm research has led to the normalisation of normative hierarchies and power asymmetries between geographies in the diffusion context. The critical norm research, complementarily, while extensively criticising such biased scholarly practices, failed to diagnose the problem as diffusionism and thus failed to benefit from the informed conclusions the anthropological schools offered regarding beyond-diffusionism in diffusion research. The paper takes this as a basis for highlighting the necessity to further extend the dialogue between IR and social sciences and humanities on issues including diffusion.

Scattering for defocusing energy subcritical nonlinear wave equations

We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the range $3 < p< 5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.

Numerical simulations for the energy-supercritical nonlinear wave equation

We carry out numerical simulations of the defocusing energy-supercritical nonlinear wave equation for a range of spherically-symmetric initial conditions. We demonstrate numerically that the critical Sobolev norm of solutions remains bounded in time. This lends support to conditional scattering results that have been recently established for nonlinear wave equations.

Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient

In this paper, we first prove the global well-posedness of 3-D anisotropic Navier-Stokes system provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data. Then we shall construct a family of initial data, u0,ν, which vary fast enough in the vertical variable and which can be arbitrarily large in the space BMO−1. Yet u0,ν still generates a unique global solution to the classical 3-D Navier-Stokes system provided that ν is sufficiently large.

Politicized Sustainability and Agricultural Policy: Comparing Norm Understandings of International Organizations

ABSTRACT The article connects research on politicization, policy change and policy autonomy, and critical constructivist norm research. It asks how the politicized global norm cluster of sustainability impacts international organizations’ (IOs) agricultural policies. The qualitative comparative case study analysis shows that the FAO, UNEP and WTO react to politicized sustainability but pursue different norm interpretations and prioritizations of agricultural approaches. The different understandings are explained with institutional factors (policy autonomy and reflective capacities) that channel norm interpretations and prioritizations of agricultural approaches. The article concludes that politicization is necessary but not sufficient to explain IOs’ norm understandings and policies towards sustainable agriculture.

Bounded solutions to an energy subcritical non-linear wave equation on [formula omitted

In this work we consider an energy subcritical semi-linear wave equation (3<p<5){∂t2u−Δu=ϕ(x)|u|p−1u,(x,t)∈R3×R;u|t=0=u0∈H˙sp(R3);∂tu|t=0=u1∈H˙sp−1(R3); where sp=3/2−2/(p−1) and the function ϕ:R3→[−1,1] is a radial continuous function with a limit at infinity. We prove that unless the elliptic equation −ΔW=ϕ(x)|W|p−1W has a nonzero radial solution W∈C2(R3)∩H˙sp(R3), any radial solution u with a finite uniform upper bound on the critical Sobolev norm ‖(u(⋅,t),∂tu(⋅,t))‖H˙sp×H˙sp(R3) for all t in the maximal lifespan must be a global solution in time and scatter.

The global well-posedness and scattering for the 5-dimensional defocusing conformal invariant NLW with radial initial data in a critical Besov space

In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space $\dot{B}^3_{1,1}\times\dot{B}^2_{1,1}(\mathbb{R}^5)$. This is the five dimensional analogue of \cite{dodson-2016}, which is the first result on the global well-posedness and scattering of the energy subcritical nonlinear wave equation without the uniform boundedness assumption on the critical Sobolev norms employed as a substitute of the missing conservation law with respect to the scaling invariance of the equation. The proof is based on exploiting the structure of the radial solution, developing the Strichartz-type estimates and incorporation of the strategy in \cite{dodson-2016}, where we also avoid a logarithm-type loss by employing the inhomogeneous Strichartz estimates.

Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

This paper is concerned with two dual aspects of the regularity question for the Navier–Stokes equations. First, we prove a local-in-time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space $$L^3$$, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and Sverak, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if $$(0, T^*)$$ is a singular point, then $$\begin{aligned} \Vert u(\cdot ,t)\Vert _{L^{3}(B_{R}(0))}\geqq \gamma _{univ},\qquad R=O(\sqrt{T^*-t}). \end{aligned}$$This result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely $$L^{3,\infty }$$ and the Besov space $${\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }$$, $$p\in (3,\infty )$$.