ODE Arguments Research Articles

Rigidity of critical metrics for quadratic curvature functionals

In this paper we prove new rigidity results for complete, possibly non-compact, critical metrics of the quadratic curvature functionals Ft2=∫|Ricg|2dVg+t∫Rg2dVg, t∈R, and S2=∫Rg2dVg. We show that (i) flat surfaces are the only critical points of S2, (ii) flat three-dimensional manifolds are the only critical points of Ft2 for every t>−13, (iii) three-dimensional scalar flat manifolds are the only critical points of S2 with finite energy and (iv) n-dimensional, n>4, scalar flat manifolds are the only critical points of S2 with finite energy and scalar curvature bounded below. In case (i), our proof relies on rigidity results for conformal vector fields and an ODE argument; in case (ii) we draw upon some ideas of M.T. Anderson concerning regularity, convergence and rigidity of critical metrics; in cases (iii) and (iv) the proofs are self-contained and depend on new pointwise and integral estimates.

Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with χ 3 nonlinear response

We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-type identities.

Sharp concentration estimates near criticality for radial sign‐changing solutions of Dirichlet and Neumann problems

We consider radial solutions of the slightly subcritical problem − Δ u ε = | u ε | 4 n − 2 − ε u ε either on R n ( n ⩾ 3 ) or in a ball B satisfying Dirichlet or Neumann boundary conditions. In particular, we provide sharp rates and constants describing the asymptotic behavior (as ε → 0 ) of all local minima and maxima of u ε and of the value of the derivative u ε ′ at the zeros of the solution. Our proof is done by induction and uses energy estimates, blow-up/normalization techniques, a radial pointwise Pohozaev identity, and some ODE arguments. As corollaries, we complement a known asymptotic approximation of the Dirichlet nodal solution in terms of a tower of bubbles and present a similar formula for the Neumann problem.

Estimates of Lifespan and Blow-up Rates for the Wave Equation with a Time-dependent Damping and a Power-type Nonlinearity

We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.

Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential

We study the Cauchy problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.

$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $\beta = (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.

Existence and stability of planar shocks of viscous scalar conservation laws with space-periodic flux

The goal of this paper is to prove the existence and stability of shocks for viscous scalar conservation laws with space periodic flux, in the multi-dimensional case. Such a result had been proved by the first author in one space dimension, but the extension to a multi-dimensional setting makes the existence proof non-trivial. We construct approximate solutions by restricting the size of the domain and then passing to the limit as the size of the domain goes to infinity. One of the key steps is a “normalization” procedure, which ensures that the limit objects obtained by the approximation scheme are indeed shocks. The proofs rely on elliptic PDE theory rather than ODE arguments as in the 1d case. Once the existence of shocks is proved, their stability follows from classical arguments based on the theory of dynamical systems.

A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation

In this paper, we are concerned with radially symmetric solutions with a vortex to a nonlinear Schrödinger equation:-ħ$^2 \Delta v+($ ħ$^2\omega^2/|x|^2+V(x))v=f(v)$ in $\mathbf R^2.$We give precise asymptotic profiles of solutions as ħ$\rightarrow 0$ by variational methods and ODE arguments.

Myth and thepolisin Bacchylides' Eleventh Ode

AbstractBacchylides' eleventh epinician ends not with renewed praise of the victor but with an extension to the ode's main myth which forges a link between the Arcadian cult of Artemis founded by Proetus and his daughters and the foundation of the victor's home city of Metapontum by Achaean colonists identified with the heroic captors of Troy. The culmination of the ode in praise of a successful colonial foundation, it is argued, is the key to the principles on which Bacchylides has selected and moulded the mythological elements that he deploys in the rest of the ode. Proetus' foundation of Tiryns resolves the civil strife which threatened to destroy Argos and commends colonization as a means of social and political progress; while the cult of Artemis at Lousoi, founded to expiate the Proetids' offence against Hera, emphasizes the role of marriage in maintaining the strength and solidarity of the community. The emphasis in both these myths on the divine intervention which rectifies human error links them to the experience of the victor and to the theme of the proper cultivation of the gods that is emphasized in the ode's conclusion. In constructing mythical narratives that are exemplary for the victor and his community, Bacchylides departs from mythological tradition in significant respects and in ways which suggest that the ode's argument reflects both the victor's status in the community and perhaps also the circumstances of its own performance.

Coarsening rates for models of multicomponent phase separation

We study the coarsening of solutions of two models of multicomponent phase separation. One is a constant mobility system; the other is a degenerate mobility system. These models are natural generalizations of the Cahn–Hilliard equation to the case of a vector-valued order parameter. It has been conjectured that the characteristic length scale `(t) grows like t1/3 as t →∞ for the first case and ` ∼ t1/4 for the second case. We prove a weak one-sided version of this assertion. Our method follows a strategy introduced by Kohn and Otto for problems with a scalar-valued order parameter; it combines a dissipation relationship with an isoperimetric inequality and an ODE argument. We also address a related model for anisotropic epitaxial growth.

Upper Bounds on Coarsening Rates

We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as \(\) in the first case and \(\) in the second. We prove a weak, one-sided version of this assertion – showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.

Nostalgia's Freight in Wordsworth's Intimations Ode

essentially nostalgic text. This view of the poem has contended that the poet's philosophic mind synthesizes objective and subjective orders to produce either Cleanth Brooks's paradox of the or Jerome McCann's Romantic ideology.' The problem with this theoretical binary of imagination construed as revelation or false consciousness is that it tends to exclude the possibility that a poem might enact the critical distance poets require to break the usual (or, in the language of Romanticism, customary) correspondences between inner and outer worlds that comprise the structures by which we live. Without sufficiently acknowledging the resources available to poetic and cultural agency even within apparently constraining structure, readers are quite literally bound to confuse Wordsworth's nostalgic material with poetic practice. In brief, the perception that the Ode promotes a politics of nostalgia may indicate a premise about agency shared by otherwise contrary critical methods. Against the tradition of readings that assume nostalgia, then judge it as cure or regressive ideal, I argue that it serves as a strategic medium that undermines nostalgia's putative status as the revered (or reviled) basis for the Ode's argument about recollection and identity.2 As Kenneth Johnston has argued, Wordsworth aims to use mental custom against itself, a goal he images in the Fenwick note as an Archimedean fulcrum point whereon to rest machine; once gained, that point empowers the poet to lever off the weight of his own mind.3 I suggest that killing weight leveraged breeds joy because the poet has gained agency from the customary formalism that is nostalgia's inertial freight. The theoretical dimension of my argument is adapted from the work of Pierre Bourdieu and Anthony Giddens, which conceptualizes practice such that it enables agency even within structural constraints that seem to demand strict reproduction or imitation.

Lower bounds on width of Stark resonances in one dimension

We study perturbations of one-dimensional Schr6dinger operators -(d2/dx 2) + V(x) by potentials 2 x corresponding to constant electric fields. Under such perturbations, the continuous spectrum expands to the entire real axis while the eigenvalues of the unperturbed operator disappear. The eigenvalues of the unperturbed operators turn into resonances. The main problem here is to estimate the width of these resonances. Upper bounds were given by many authors, e.g. [2, 7, 9], while lower bounds have only been obtained so far by [2, 3]. Harrell and Simon [2] obtained a lower bound for the resonance corresponding to the lowest eigenvalue of the hydrogen atom. To this end, following Schr6dinger, Oppenheimer, and Epstein, they have separated variables in parabolic coordinates, reducing the problem to a one-dimensional one, which they then transformed to the one-dimensional anharmonic oscillator for which they used the one-dimensional WKB method to derive their estimates. Harrell [ 3] used ODE arguments (in the spirit of the WKB method) to obtain general lower bounds for resonances of Schr6dinger operators in one dimension. We use ODE arguments in our study, but our approach is different from the approach of [3]. We consider general one-dimensional Schr6dinger operators and their arbitrary eigenvalues. Using elementary and rather general estimates related to the spectral deformation method, we give lower bounds for resonances corresponding to eigenvalues of the unperturbed problem.

Blow-up for self-interacting fractional Ginzburg–Landau equation

The blow-up of solutions for the Cauchy problem of fractional Ginzburg-Landau equation with non-positive nonlinearity is shown by an ODE argument. Moreover, in one dimensional case, the optimal lifespan estimate for size of initial data is obtained.