The magnetohydrodynamics system consists of the Navier-Stokes and Maxwell's equations, coupled through multiples of nonlinear terms. Such a system forced by space-time white noise has been studied by physicists for decades, and the rigorous proof of its solution theory has been recently established in Yamazaki (2019, arXiv:1910.04820 [math.AP]) using the theory of paracontrolled distributions and a technique of coupled renormalizations. When an equation is well-posed, and it is approximated by replacing the differentiation operator by reasonable discretization schemes with a parameter, it is widely believed that a solution of the approximating equation should converge to the solution of the original equation as the parameter approaches zero. We prove otherwise in the case of the three-dimensional magnetohydrodynamics system forced by space-time white noise. Specifically, it is proven that the limit of the solution to the approximating system with an additional 32 drift terms solves the original system. These 32 drift terms depend on the choice of approximations, can be calculated explicitly in the process of renormalizations, and essentially represent a spatial version of It$\hat{\mathrm{o}}$-Stratonovich correction terms. In particular, the proof relies on the technique of coupled renormalizations again, as well as taking advantage of the special structure of the magnetohydrodynamics system on many occasions.
Most recently, in arXiv:1907.05360 [math.AP], we introduced the theory of heatable currents and proved Onsager's conjecture on Riemannian manifolds with boundary, where the weak solution has $B_{3,1}^{\frac{1}{3}}$ spatial regularity. In this sequel, by applying techniques from geometric microlocal analysis to construct the Hodge-Neumann heat kernel, we obtain off-diagonal decay and local Bernstein estimates, and then use them to extend the result to the Besov space $\widehat{B}_{3,V}^{\frac{1}{3}}$, which generalizes both the space $\widehat{B}_{3,c(\mathbb{N})}^{1/3}$ from arXiv:1310.7947 [math.AP] and the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from arXiv:1902.07120 [math.AP] -- the best known function space where Onsager's conjecture holds on flat backgrounds.
We consider non-ergodic magnetic random Schrödinger operators with a bounded magnetic vector potential. We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of arguments from Klein (Commun Math Phys 323(3):1229–1246, 2013), combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from Borisov et al. (J Math Phys, arXiv:1512.06347 [math.AP]). This generalizes Klein’s result to operators with a bounded magnetic vector potential. Moreover, we study the dependence of the Wegner-constant on the disorder parameter. In particular, we show that above the model-dependent threshold $$E_0(\infty ) \in (0, \infty ]$$E0(∞)∈(0,∞], it is impossible that the Wegner-constant tends to zero if the disorder increases. This result is new even for the standard (ergodic) Anderson Hamiltonian without magnetic field.
In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper arXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics which do not admit local limiting Carleman weights at any point is residual, showing that it contains the set of metrics for which there are no local conformal diffeomorphisms between any distinct open subsets. This paper is a continuation of arXiv:1411.4887 [math.AP] in order to prove that the set of Riemannian metrics which do not admit local limiting Carleman weights \emph{at any point} is open and dense.
We prove that the Gibbs measures [Formula: see text] for a class of Hamiltonian equations written as [Formula: see text] on the real line are invariant under the flow of [Formula: see text] in the sense that there exist random variables [Formula: see text] whose laws are [Formula: see text] (thus independent from [Formula: see text]) and such that [Formula: see text] is a solution to [Formula: see text]. Besides, for all [Formula: see text], [Formula: see text] is almost surely not in [Formula: see text] which provides as a direct consequence the existence of global weak solutions for initial data not in [Formula: see text]. The proof uses Prokhorov’s theorem, Skorohod’s theorem, as in the strategy in [N. Burq, L. Thomann and N. Tzvetkov, Remarks on the Gibbs measures for nonlinear dispersive equations, preprint (2014); arXiv:1412.7499v1 [math.AP]] and Feynman–Kac’s integrals.
This article describes the PRIMME software package for solving large, sparse Hermitian standard eigenvalue problems. The difficulty and importance of these problems have increased over the years, necessitating the use of preconditioning and near optimally converging iterative methods. However, the complexity of tuning or even using such methods has kept them outside the reach of many users. Responding to this problem, we have developed PRIMME, a comprehensive package that brings state-of-the-art methods from “bleeding edge” to production, with the best possible robustness, efficiency, and a flexible, yet highly usable interface that requires minimal or no tuning. We describe (1) the PRIMME multimethod framework that implements a variety of algorithms, including the near optimal methods GD+kand JDQMR; (2) a host of algorithmic innovations and implementation techniques that endow the software with its robustness and efficiency; (3) a multilayer interface that captures our experience and addresses the needs of both expert and end users.
EQUIVARIANT SCHR ODINGER MAPS IN TWO SPATIAL DIMENSIONS: THE H 2 TARGET arXiv:1212.2566v1 [math.AP] 11 Dec 2012 I. BEJENARU, A. IONESCU, C. KENIG, AND D. TATARU Abstract. We consider equivariant solutions for the Schr¨odinger map problem from R 2+1 to H 2 with finite energy and show that they are global in time and scatter. 1. Introduction The Schr¨odinger map equation in R 2+1 with values into S µ ⊂ R 3 is given by u t = u × µ ∆u, u(0) = u 0 where µ = ±1, the connected Riemannian manifolds S µ , S 1 = S 2 = {y = (y 0 , y 1 , y 2 ) ∈ R 3 : y 1 2 + y 2 2 + y 3 2 = 1}; S −1 = H 2 = {y = (y 0 , y 1 , y 2 ) ∈ R 3 : −y 1 2 − y 2 2 + y 3 2 = 1, y 3 > 0}, with the Riemannian structures induced by the Euclidean metric g 1 = dy 0 2 + dy 1 2 + dy 2 2 on S 1 , respectively the Minkowski metric g −1 = −dy 0 2 + dy 1 2 + dy 2 2 on S −1 . Thus S 1 is the 2-dimensional sphere S 2 , while S −1 is the 2-dimensional hyperbolic space H 2 . With η µ = diag(1, 1, µ), the cross product × µ is defined by v × µ w := η µ · (v × w). This equation admits a conserved energy, Z E(u) = |∇u| 2 µ dx 2 R 2 and is invariant with respect to the dimensionless scaling u(t, x) → u(λ 2 t, λx). The energy is invariant with respect to the above scaling, therefore the Schr¨odinger map equation in R 2+1 is energy critical. The local theory for classical data was established in [25] and [21]. We recall Theorem 1.1 (McGahagan). If u 0 ∈ H ˙ 1 ∩ H ˙ 3 then there exists a time T > 0, such that (1.1) has a unique solution in L ∞ t ([0, T ] : H ∩ H ). The local and global in time of the Schr¨odinger map problem with small data has been intensely studied for the case µ = 1 corresponding to S 2 as target, see [3], [4], [5], [6], [9], [15], [16]. The state of the art result for the problem with small data was established by the authors in [6] where they proved that classical solutions (and in fact rough solutions too) with small energy are global in time. These results are expected to extend to the case µ = −1, corresponding to H 2 as a target. I.B. was supported in part by NSF grant DMS-1001676. A. I. was partially supported by a Packard Fellowship and NSF grant DMS-1065710. C.K. was supported in part by NSF grant DMS-0968742. D.T. was supported in part by the Miller Foundation and by NSF grant DMS-0801261.
We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation utt − uxx + u − |u|2αu = 0 on the line. Using mixed numerical and analytical methods we find that solutions starting from even initial data, fine-tuned to the threshold, are trapped by the static solution S for intermediate times. The details of trapping are shown to depend on the power α, namely, we observe fast convergence to S for α > 1, slow convergence for α = 1, and very slow (if any) convergence for 0 2) by Krieger, Nakanishi, and Schlag [“Global dynamics above from the ground state energy for the one-dimensional NLKG equation,” preprint arXiv:1011.1776 [math.AP]].We study dynamics near the threshold for blowup in the focusing nonlinear Klein-Gordon equation utt − uxx + u − |u|2αu = 0 on the line. Using mixed numerical and analytical methods we find that solutions starting from even initial data, fine-tuned to the threshold, are trapped by the static solution S for intermediate times. The details of trapping are shown to depend on the power α, namely, we observe fast convergence to S for α > 1, slow convergence for α = 1, and very slow (if any) convergence for 0 2) by Krieger, Nakanishi, and Schlag [“Global dynamics above from the ground state energy for the one-dimensional NLKG equation,” preprint arXiv:1011.1776 [math.AP]].
Abstract We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ( α 1 / 2 ) dissipation ( − Δ ) α . This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ( α = 1 / 2 ) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L 2 to L ∞ , from L ∞ to Holder ( C δ , δ > 0 ), and from Holder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L ∞ , but it does not appear that their approach can be easily extended to establish the Holder continuity of L ∞ solutions. In order for their approach to work, we require the velocity to be in the Holder space C 1 − 2 α . Higher regularity starting from C δ with δ > 1 − 2 α can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Holder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincare Anal. Non Lineaire, in press].
This paper focuses on the study of existence and uniqueness of distributional and classical solutions to the Cauchy Boltzmann problem for the soft potential case assuming Sn−1 integrability of the angular part of the collision kernel (Grad cut-off assumption). For this purpose we revisit the Kaniel–Shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local Maxwellian regime with possibly infinite initial mass. We study the propagation of regularity using a recent estimate for the positive collision operator given in (Alonso et al. in Convolution inequalities for the Boltzmann collision operator. arXiv:0902.0507 [math.AP]) , by E. Carneiro and the authors, that allows us to show such propagation without additional conditions on the collision kernel. Finally, an Lp-stability result (with 1≤p≤∞) is presented assuming the aforementioned condition.
We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]
Abstract We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schrodinger and Gross–Pitaevskii equations on the exterior of a nontrapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate [R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrodinger equation on domains, arxiv: math.AP/0512639 , Bull. Soc. Math. France, submitted for publication] with a smoothing effect on exterior domains [N. Burq, P. Gerard, N. Tzvetkov, On nonlinear Schrodinger equations in exterior domains, Ann. I.H.P. (2004) 295–318].