Given a suitable solution
We study the behavior of solutions to the incompressible 2
 Publication Date:
 NSFPAR ID:
 10394858
 Journal Name:
 Archive for Rational Mechanics and Analysis
 Volume:
 247
 Issue:
 1
 ISSN:
 00039527
 Publisher:
 Springer Science + Business Media
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract V (t ,x ) to the Korteweg–de Vries equation on the real line, we prove global wellposedness for initial data . Our conditions on$$u(0,x) \in V(0,x) + H^{1}(\mathbb {R})$$ $u(0,x)\in V(0,x)+{H}^{1}\left(R\right)$V do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles satisfy our hypotheses. In particular, we can treat localized perturbations of the muchstudied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022.$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$ $V(0,x)\in {H}^{5}(R/Z)$https://doi.org/10.1088/13616544/ac37f5 ) we show that smooth steplike initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019.https://doi.org/10.4007/annals.2019.190.1.4 ) where . In that setting, it is known that$$V\equiv 0$$ $V\equiv 0$ is sharp in the class of$$H^{1}(\mathbb {R})$$ ${H}^{1}\left(R\right)$ spaces.$$H^s(\mathbb {R})$$ ${H}^{s}\left(R\right)$ 
Abstract A longstanding problem in mathematical physics is the rigorous derivation of the incompressible Euler equation from Newtonian mechanics. Recently, HanKwan and Iacobelli (Proc Am Math Soc 149:3045–3061, 2021) showed that in the monokinetic regime, one can directly obtain the Euler equation from a system of
N particles interacting in ,$${\mathbb {T}}^d$$ ${T}^{d}$ , via Newton’s second law through a$$d\ge 2$$ $d\ge 2$supercritical meanfield limit . Namely, the coupling constant in front of the pair potential, which is Coulombic, scales like$$\lambda $$ $\lambda $ for some$$N^{\theta }$$ ${N}^{\theta}$ , in contrast to the usual meanfield scaling$$\theta \in (0,1)$$ $\theta \in (0,1)$ . Assuming$$\lambda \sim N^{1}$$ $\lambda \sim {N}^{1}$ , they showed that the empirical measure of the system is effectively described by the solution to the Euler equation as$$\theta \in (1\frac{2}{d(d+1)},1)$$ $\theta \in (1\frac{2}{d(d+1)},1)$ . HanKwan and Iacobelli asked if their range for$$N\rightarrow \infty $$ $N\to \infty $ was optimal. We answer this question in the negative by showing the validity of the incompressible Euler equation in the limit$$\theta $$ $\theta $ for$$N\rightarrow \infty $$ $N\to \infty $ . Our proof is based on Serfaty’s modulatedenergy method, but compared to that of HanKwan and Iacobelli, crucially uses an improved “renormalized commutator” estimate to obtain the larger range for$$\theta \in (1\frac{2}{d},1)$$ $\theta \in (1\frac{2}{d},1)$ . Additionally, we show that for$$\theta $$ $\theta $ , one cannot, in general, expect convergence in the modulated energy notion of distance.$$\theta \le 1\frac{2}{d}$$ $\theta \le 1\frac{2}{d}$ 
Abstract In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying
, for any even integer$$K^2 = 4p_g8$$ ${K}^{2}=4{p}_{g}8$ . These surfaces also have unbounded irregularity$$p_g\ge 4$$ ${p}_{g}\ge 4$q . We carry out our study by investigating the deformations of the canonical morphism , where$$\varphi :X\rightarrow {\mathbb {P}}^N$$ $\phi :X\to {P}^{N}$ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):54895507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of$$\varphi $$ $\phi $ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of$$\varphi $$ $\phi $ is twotoone onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of$$\varphi $$ $\phi $X are unobstructed even though does not vanish. Consequently,$$H^2(T_X)$$ ${H}^{2}\left({T}_{X}\right)$X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we showmore » 
Abstract The free multiplicative Brownian motion
is the large$$b_{t}$$ ${b}_{t}$N limit of the Brownian motion on in the sense of$$\mathsf {GL}(N;\mathbb {C}),$$ $\mathrm{GL}(N\u037eC),$ distributions. The natural candidate for the large$$*$$ $\ast $N limit of the empirical distribution of eigenvalues is thus the Brown measure of . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$b_{t}$$ ${b}_{t}$ that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$\Sigma _{t}$$ ${\Sigma}_{t}$ on$$W_{t}$$ ${W}_{t}$ which is strictly positive and real analytic on$$\overline{\Sigma }_{t},$$ ${\overline{\Sigma}}_{t},$ . This density has a simple form in polar coordinates:$$\Sigma _{t}$$ ${\Sigma}_{t}$ where$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ $\begin{array}{c}{W}_{t}(r,\theta )=\frac{1}{{r}^{2}}{w}_{t}\left(\theta \right),\end{array}$ is an analytic function determined by the geometry of the region$$w_{t}$$ ${w}_{t}$ . We show also that the spectral measure of free unitary Brownian motion$$\Sigma _{t}$$ ${\Sigma}_{t}$ is a “shadow” of the Brown measure of$$u_{t}$$ ${u}_{t}$ , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.$$b_{t}$$ ${b}_{t}$ 
Abstract We present a proof of concept for a spectrally selective thermal midIR source based on nanopatterned graphene (NPG) with a typical mobility of CVDgrown graphene (up to 3000
), ensuring scalability to large areas. For that, we solve the electrostatic problem of a conducting hyperboloid with an elliptical wormhole in the presence of an$$\hbox {cm}^2\,\hbox {V}^{1}\,\hbox {s}^{1}$$ ${\text{cm}}^{2}\phantom{\rule{0ex}{0ex}}{\text{V}}^{1}\phantom{\rule{0ex}{0ex}}{\text{s}}^{1}$inplane electric field. The localized surface plasmons (LSPs) on the NPG sheet, partially hybridized with graphene phonons and surface phonons of the neighboring materials, allow for the control and tuning of the thermal emission spectrum in the wavelength regime from to 12$$\lambda =3$$ $\lambda =3$ m by adjusting the size of and distance between the circular holes in a hexagonal or square lattice structure. Most importantly, the LSPs along with an optical cavity increase the emittance of graphene from about 2.3% for pristine graphene to 80% for NPG, thereby outperforming stateoftheart pristine graphene light sources operating in the nearinfrared by at least a factor of 100. According to our COMSOL calculations, a maximum emission power per area of$$\upmu$$ $\mu $ W/$$11\times 10^3$$ $11\times {10}^{3}$ at$$\hbox {m}^2$$ ${\text{m}}^{2}$ K for a bias voltage of$$T=2000$$ $T=2000$ V is achieved by controlling the temperature of the hot electrons through the Joule heating. By generalizing Planck’s theory to any grey body and derivingmore »$$V=23$$ $V=23$