## Recursos de colección

Analysis & PDE

1. #### Scale-free unique continuation principle for spectral projectors, eigenvalue-lifting and Wegner estimates for random Schrödinger operators

Nakić, Ivica; Täufer, Matthias; Tautenhahn, Martin; Veselić, Ivan
We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector [math] of a Schrödinger operator [math] on a cube of side [math] , with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors [math] with small [math] . Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. Our main application of such an estimate is to find lower bounds for the lifting of eigenvalues under semidefinite positive perturbations, which in turn can be applied to derive a Wegner estimate for...

2. #### Global weak solutions for generalized SQG in bounded domains

Nguyen, Huy Quang
We prove the existence of global [math] weak solutions for a family of generalized inviscid surface quasigeostrophic (SQG) equations in bounded domains of [math] . In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded domains.

3. #### Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations

Duyckaerts, Thomas; Yang, Jianwei
We consider a wave equation in three space dimensions, with a power-like nonlinearity which is either focusing or defocusing. The exponent is greater than 3 (conformally supercritical) and not equal to 5 (not energy-critical). We prove that for any radial solution which does not scatter to a linear solution, an adapted scale-invariant Sobolev norm goes to infinity at the maximal time of existence. The proof uses a conserved generalized energy for the radial linear wave equation, new Strichartz estimates adapted to this generalized energy, and a bound from below of the generalized energy of any nonzero solution outside wave cones....

4. #### Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains

Bonforte, Matteo; Figalli, Alessio; Vázquez, Juan Luis
We provide a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form [math] , [math] , where the operator [math] belongs to a general class of linear operators, and the equation is posed in a bounded domain [math] . As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, [math] can be a fractional power of a uniformly elliptic operator with [math] coefficients. Since the nonlinearity is given by [math] with [math]...

5. #### On a bilinear Strichartz estimate on irrational tori

Fan, Chenjie; Staffilani, Gigliola; Wang, Hong; Wilson, Bobby
We prove a bilinear Strichartz-type estimate for irrational tori via a decoupling-type argument, as used by Bourgain and Demeter (2015), recovering and generalizing a result of De Silva, Pavlović, Staffilani and Tzirakis (2007). As a corollary, we derive a global well-posedness result for the cubic defocusing NLS on two-dimensional irrational tori with data of infinite energy.

6. #### Beyond the BKM criterion for the 2D resistive magnetohydrodynamic equations

Agélas, Léo
The question of whether the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only magnetic diffusion can develop a finite-time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion less restrictive than the Beale–Kato–Majda (BKM) regularity criterion type, namely any solution [math] with [math] remains in [math] up to time [math] under the assumption that ¶ $∫ 0 T ∥ ∇ u ( t ) ∥ ∞ 1 2 log ( e + ∥ ∇ u ( t ) ∥ ∞ ) d t < + ∞ .$ ¶ This regularity criterion may stand as a great improvement...

7. #### On rank-2 Toda systems with arbitrary singularities: local mass and new estimates

Lin, Chang-Shou; Wei, Jun-cheng; Yang, Wen; Zhang, Lei
For all rank-2 Toda systems with an arbitrary singular source, we use a unified approach to prove:

1. The pair of local masses [math] at each blowup point has the expression [math] where [math] , [math] , [math] .
2. At each vortex point [math] if [math] are integers and [math] , then all the solutions of Toda systems are uniformly bounded.
3. If the blowup point [math] is a vortex point [math] and [math] and [math] are linearly independent over [math] , then [math]
¶ The Harnack-type inequalities of 3 are important for studying the bubbling behavior near...

8. #### Applications of small-scale quantum ergodicity in nodal sets

Hezari, Hamid
The goal of this article is to draw new applications of small-scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius [math] then one can achieve improvements on the recent upper bounds of Logunov (2016) and Logunov and Malinnikova (2016) on the size of nodal sets, according to a certain power of [math] . We also show that the doubling estimates and the order-of-vanishing results of Donnelly and Fefferman (1988, 1990) can be improved. Due to results of Han (2015) and Hezari and Rivière (2016), small-scale QE holds on negatively...

9. #### $C^1$ regularity of orthotropic $p$-harmonic functions in the plane

Bousquet, Pierre; Brasco, Lorenzo
We prove that local weak solutions of the orthotropic [math] -harmonic equation in [math] are [math] functions.

10. #### 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...

11. #### 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.

12. #### 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...

13. #### 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...

14. #### 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.

15. #### 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] .

16. #### 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.

17. #### 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.

18. #### 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.

19. #### 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...

20. #### 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.

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