## Recursos de colección

Analysis & PDE

1. #### Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation

Tao, Terence
Let [math] . We consider the global Cauchy problem for the generalized Navier–Stokes system ¶ $∂ t u + ( u ⋅ ∇ ) u = − D 2 u − ∇ p , ∇ ⋅ u = 0 , u ( 0 , x ) = u 0 ( x )$ ¶ for [math] and [math] , where [math] is smooth and divergence free, and [math] is a Fourier multiplier whose symbol [math] is nonnegative; the case [math] is essentially Navier–Stokes. It is folklore that one has global regularity in the critical and subcritical hyperdissipation regimes [math] for [math] . We improve this slightly by establishing global regularity under the slightly weaker condition...

2. #### Stability for strongly coupled critical elliptic systems in a fully inhomogeneous medium

Druet, Olivier; Hebey, Emmanuel
We investigate and prove analytic stability for strongly coupled critical elliptic systems in the inhomogeneous context of a compact Riemannian manifold.

3. #### Periodic stochastic Korteweg–de Vries equation with additive space-time white noise

We prove the local well-posedness of the periodic stochastic Korteweg–de Vries equation with the additive space-time white noise. To treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space [math] for [math] , [math] such that [math] . In establishing local well-posedness, we use a variant of the Bourgain space adapted to [math] and establish a nonlinear estimate on the second iteration on the integral formulation. The deterministic part of the nonlinear estimate also yields the local well-posedness of the deterministic KdV in [math] , the space of finite Borel measures on...

4. #### Global existence of smooth solutions of a 3D log-log energy-supercritical wave equation

Roy, Tristan
We prove global existence of smooth solutions of the 3D log-log energy-supercritical wave equation ¶ $∂ t t u − △ u = − u 5 log c ( log ( 1 0 + u 2 ) )$ ¶ with [math] and smooth initial data [math] . First we control the [math] norm of the solution on an arbitrary size time interval by an expression depending on the energy and an a priori upper bound of its [math] norm, with [math] . The proof of this long time estimate relies upon the use of some potential decay estimates and a modification of an argument by Tao. Then we find an a posteriori...

5. #### The high exponent limit $p \to \infty$ for the one-dimensional nonlinear wave equation

Tao, Terence
We investigate the behaviour of solutions [math] to the one-dimensional nonlinear wave equation [math] with initial data [math] , [math] , in the high exponent limit [math] (holding [math] fixed). We show that if the initial data [math] are smooth with [math] taking values in [math] and obey a mild nondegeneracy condition, then [math] converges locally uniformly to a piecewise limit [math] taking values in the interval [math] , which can in principle be computed explicitly.

6. #### Roth's theorem in $\mathbb{Z}_4^n$

Sanders, Tom
We show that if [math] contains no three-term arithmetic progressions in which all the elements are distinct then [math] .

7. #### On the global well-posedness of the one-dimensional Schrödinger map flow

Rodnianski, Igor; Rubinstein, Yanir; Staffilani, Gigliola
We establish the global well-posedness of the initial value problem for the Schrödinger map flow for maps from the real line into Kähler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding.

8. #### Dynamics of vortices for the complex Ginzburg–Landau equation

Miot, Evelyne
We study a complex Ginzburg–Landau equation in the plane, which has the form of a Gross–Pitaevskii equation with some dissipation added. We focus on the regime corresponding to well-prepared unitary vortices and derive their asymptotic motion law.

9. #### Heat-flow monotonicity of Strichartz norms

Bennett, Jonathan; Bez, Neal; Carbery, Anthony; Hundertmark, Dirk
Our main result is that for [math] the classical Strichartz norm [math] associated to the free Schrödinger equation is nondecreasing as the initial datum [math] evolves under a certain quadratic heat flow.

10. #### Lower estimates on microstates free entropy dimension

Shlyakhtenko, Dimitri
By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain [math] -tuples [math] . In particular, we show that [math] , where [math] and [math] is the set of values of derivations [math] with the property that [math] . We show that for [math] sufficiently small (depending on [math] ) and [math] a [math] -semicircular family, [math] . In particular, for small [math] , [math] -deformed free group factors have no Cartan subalgebras. An essential tool in our analysis is a free analog...

11. #### The linear profile decomposition for the Airy equation and the existence of maximizers for the Airy Strichartz inequality

Shao, Shuanglin
We establish the linear profile decomposition for the Airy equation with complex or real initial data in [math] . As an application, we obtain a dichotomy result on the existence of maximizers for the symmetric Airy Strichartz inequality.

12. #### Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

Tao, Terence
We consider the focusing mass-critical NLS [math] in high dimensions [math] , with initial data [math] having finite mass [math] . It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class [math] , and also admits global (but not unique) weak solutions in [math] . In this paper we introduce an intermediate class of solution, which we call a semi-Strichartz class solution, for which one does have global existence and uniqueness in dimensions [math] . In dimensions [math] and assuming spherical symmetry, we also show the equivalence of the Strichartz class and...

13. #### Resonances for nonanalytic potentials

Martinez, André; Ramond, Thierry; Sjöstrand, Johannes
We consider semiclassical Schrödinger operators on [math] , with [math] potentials decaying polynomially at infinity. The usual theories of resonances do not apply in such a nonanalytic framework. Here, under some additional conditions, we show that resonances are invariantly defined up to any power of their imaginary part. The theory is based on resolvent estimates for families of approximating distorted operators with potentials that are holomorphic in narrow complex sectors around [math] .

14. #### Uniqueness of ground states for pseudorelativistic Hartree equations

Lenzmann, Enno
We prove uniqueness of ground states [math] for the pseudorelativistic Hartree equation, ¶ $− Δ + m 2 Q − ( x − 1 ∗ Q 2 ) Q = − μ Q ,$ ¶ in the regime of [math] with sufficiently small [math] -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for [math] except for at most countably many [math] . ¶ Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartree-type equation (also known as the Choquard–Pekard or Schrödinger–Newton equation). Uniqueness of ground states for this...

15. #### An improved lower bound on the size of Kakeya sets over finite fields

In a recent breakthrough, Dvir showed that every Kakeya set in [math] must have cardinality at least [math] , where [math] . We improve this lower bound to [math] for a constant [math] . This pins down the correct growth of the constant [math] as a function of [math] (up to the determination of [math] ).

16. #### The pseudospectrum of systems of semiclassical operators

Dencker, Nils
The pseudospectrum (or spectral instability) of non-self-adjoint operators is a topic of current interest in applied mathematics. In fact, for non-self-adjoint operators the resolvent could be very large outside the spectrum, making numerical computation of the complex eigenvalues very hard. This has importance, for example, in quantum mechanics, random matrix theory and fluid dynamics. ¶ The occurrence of false eigenvalues (or pseudospectrum) of non-self-adjoint semiclassical differential operators is due to the existence of quasimodes, that is, approximate local solutions to the eigenvalue problem. For scalar operators, the quasimodes appear generically since the bracket condition on the principal symbol is not satisfied...

17. #### Dynamics of nonlinear Schrödinger/Gross–Pitaevskii equations: mass transfer in systems with solitons and degenerate neutral modes

Zhou, Gang; Weinstein, Michael
Nonlinear Schrödinger/Gross–Pitaevskii equations play a central role in the understanding of nonlinear optical and macroscopic quantum systems. The large time dynamics of such systems is governed by interactions of the nonlinear ground state manifold, discrete neutral modes (“excited states”) and dispersive radiation. Systems with symmetry, in spatial dimensions larger than one, typically have degenerate neutral modes. Thus, we study the large time dynamics of systems with degenerate neutral modes. This requires a new normal form (nonlinear matrix Fermi Golden Rule) governing the system’s large time asymptotic relaxation to the ground state (soliton) manifold.

18. #### The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher

Killip, Rowan; Visan, Monica; Zhang, Xiaoyi
We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation [math] for large spherically symmetric [math] initial data in dimensions [math] . In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.

19. #### CR-invariants and the scattering operator for complex manifolds with boundary

Hislop, Peter; Perry, Peter; Tang, Siu-Hung
Suppose that [math] is a strictly pseudoconvex CR manifold bounding a compact complex manifold [math] of complex dimension [math] . Under appropriate geometric conditions on [math] , the manifold [math] admits an approximate Kähler–Einstein metric [math] which makes the interior of [math] a complete Riemannian manifold. We identify certain residues of the scattering operator on [math] as conformally covariant differential operators on [math] and obtain the CR [math] -curvature of [math] from the scattering operator as well. In order to construct the Kähler–Einstein metric on [math] , we construct a global approximate solution of the complex Monge–Ampère equation on [math]...

20. #### Microlocal propagation near radial points and scattering for symbolic potentials of order zero

Hassell, Andrew; Melrose, Richard; Vasy, András
In this paper, the scattering and spectral theory of [math] is developed, where [math] is the Laplacian with respect to a scattering metric [math] on a compact manifold [math] with boundary and [math] is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system. In this case the radial points correspond precisely to critical points...

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