skip to main content

Title: Global Well-Posedness for $$H^{-1}(\mathbb {R})$$ Perturbations of KdV with Exotic Spatial Asymptotics

Given a suitable solutionV(tx) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data$$u(0,x) \in V(0,x) + H^{-1}(\mathbb {R})$$u(0,x)V(0,x)+H-1(R). Our conditions onVdo include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles$$V(0,x)\in H^5(\mathbb {R}/\mathbb {Z})$$V(0,x)H5(R/Z)satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022. we show that smooth step-like 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. where$$V\equiv 0$$V0. In that setting, it is known that$$H^{-1}(\mathbb {R})$$H-1(R)is sharp in the class of$$H^s(\mathbb {R})$$Hs(R)spaces.

more » « less
Award ID(s):
Author(s) / Creator(s):
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Communications in Mathematical Physics
Page Range / eLocation ID:
p. 1387-1439
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$Lβ,γ=-divDd+1+γ-nassociated to a domain$$\Omega \subset {\mathbb {R}}^n$$ΩRnwith a uniformly rectifiable boundary$$\Gamma $$Γof dimension$$d < n-1$$d<n-1, the now usual distance to the boundary$$D = D_\beta $$D=Dβgiven by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$Dβ(X)-β=Γ|X-y|-d-βdσ(y)for$$X \in \Omega $$XΩ, where$$\beta >0$$β>0and$$\gamma \in (-1,1)$$γ(-1,1). In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$Lβ,γ, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$D1-γ, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$|D(ln(GD1-γ))|2satisfies a Carleson measure estimate on$$\Omega $$Ω. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).

    more » « less
  2. Abstract

    In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb–Thirring constant when the eigenvalues of a Schrödinger operator$$-\Delta +V(x)$$-Δ+V(x)are raised to the power$$\kappa $$κis never given by the one-bound state case when$$\kappa >\max (0,2-d/2)$$κ>max(0,2-d/2)in space dimension$$d\ge 1$$d1. When in addition$$\kappa \ge 1$$κ1we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo–Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb–Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrödinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. In a different but related direction, we also show that the cubic nonlinear Schrödinger equation admits no orthonormal ground state in 1D, for more than one function.

    more » « less
  3. Abstract

    Let$$X\rightarrow {{\mathbb {P}}}^1$$XP1be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$ωiof Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$ωi. Given the corresponding sequence$$\Xi _i$$Ξiof Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$Ξi|Econverges to a flat connection$$A_0$$A0. Furthermore, if the restriction$$V|_E$$V|Eis of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$j=1nOE(qj-0)forndistinct points$$q_j\in E$$qjE, then these points uniquely determine$$A_0$$A0.

    more » « less
  4. Abstract

    We explore properties of the family sizes arising in a linear birth process with immigration (BI). In particular, we study the correlation of the number of families observed during consecutive disjoint intervals of time. LettingS(ab) be the number of families observed in (ab), we study the expected sample variance and its asymptotics forpconsecutive sequential samples$$S_p =(S(t_0,t_1),\dots , S(t_{p-1},t_p))$$Sp=(S(t0,t1),,S(tp-1,tp)), for$$0=t_00=t0<t1<<tp. By conditioning on the sizes of the samples, we provide a connection between$$S_p$$Spandpsequential samples of sizes$$n_1,n_2,\dots ,n_p$$n1,n2,,np, drawn from a single run of a Chinese Restaurant Process. Properties of the latter were studied in da Silva et al. (Bernoulli 29:1166–1194, 2023. We show how the continuous-time framework helps to make asymptotic calculations easier than its discrete-time counterpart. As an application, for a specific choice of$$t_1,t_2,\dots , t_p$$t1,t2,,tp, where the lengths of intervals are logarithmically equal, we revisit Fisher’s 1943 multi-sampling problem and give another explanation of what Fisher’s model could have meant in the world of sequential samples drawn from a BI process.

    more » « less
  5. Abstract

    We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$Quasi-NP=NTIME[n(logn)O(1)]and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathcal { C}$C, by showing that$\mathcal { C}$Cadmits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class${\mathcal C}$C. Say that a symmetric Boolean functionf(x1,…,xn) issparseif it outputs 1 onO(1) values of${\sum }_{i} x_{i}$ixi. We show that for every sparsef, and for all “typical”$\mathcal { C}$C, faster#SAT algorithms for$\mathcal { C}$Ccircuits imply lower bounds against the circuit class$f \circ \mathcal { C}$fC, which may bestrongerthan$\mathcal { C}$Citself. In particular:

    #SAT algorithms fornk-size$\mathcal { C}$C-circuits running in 2n/nktime (for allk) implyNEXPdoes not have$(f \circ \mathcal { C})$(fC)-circuits of polynomial size.

    #SAT algorithms for$2^{n^{{\varepsilon }}}$2nε-size$\mathcal { C}$C-circuits running in$2^{n-n^{{\varepsilon }}}$2nnεtime (for someε> 0) implyQuasi-NPdoes not have$(f \circ \mathcal { C})$(fC)-circuits of polynomial size.

    Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJACC0THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC0[Williams JACM’14] andACC0THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

    more » « less