## Recursos de colección

Analysis & PDE

1. #### Square function estimates for the Bochner–Riesz means

Lee, Sanghyuk
We consider the square-function (known as Stein’s square function) estimate associated with the Bochner–Riesz means. The previously known range of the sharp estimate is improved. Our results are based on vector-valued extensions of Bennett, Carbery and Tao’s multilinear (adjoint) restriction estimate and an adaptation of an induction argument due to Bourgain and Guth. Unlike the previous work by Bourgain and Guth on [math] boundedness of the Bochner–Riesz means in which oscillatory operators associated to the kernel were studied, we take more direct approach by working on the Fourier transform side. This enables us to obtain the correct order of smoothing,...

2. #### Dini and Schauder estimates for nonlocal fully nonlinear parabolic equations with drifts

Dong, Hongjie; Jin, Tianling; Zhang, Hong
We obtain Dini- and Schauder-type estimates for concave fully nonlinear nonlocal parabolic equations of order [math] with rough and nonsymmetric kernels and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini-type estimates in the spacial variable. This is a continuation of work by the authors Dong and Zhang.

3. #### Dolgopyat's method and the fractal uncertainty principle

Dyatlov, Semyon; Jin, Long
We show a fractal uncertainty principle with exponent [math] , [math] , for Ahlfors–David regular subsets of [math] of dimension [math] . This is an improvement over the volume bound [math] , and [math] is estimated explicitly in terms of the regularity constant of the set. The proof uses a version of techniques originating in the works of Dolgopyat, Naud, and Stoyanov on spectral radii of transfer operators. Here the group invariance of the set is replaced by its fractal structure. As an application, we quantify the result of Naud on spectral gaps for convex cocompact hyperbolic surfaces and obtain...

4. #### Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces

Albritton, Dallas
We obtain an improved blow-up criterion for solutions of the Navier–Stokes equations in critical Besov spaces. If a mild solution [math] has maximal existence time [math] , then the non-endpoint critical Besov norms must become infinite at the blow-up time: ¶ $lim t ↑ T ∗ ∥ u ( ⋅ , t ) ∥ Ḃ p , q − 1 + 3 ∕ p ( ℝ 3 ) = ∞ , 3 < p , q < ∞ .$ ¶ In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply...

5. #### The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

Stefanov, Plamen; Yang, Yang
We consider the Dirichlet-to-Neumann map [math] on a cylinder-like Lorentzian manifold related to the wave equation related to the metric [math] , the magnetic field [math] and the potential [math] . We show that we can recover the jet of [math] on the boundary from [math] up to a gauge transformation in a stable way. We also show that [math] recovers the following three invariants in a stable way: the lens relation of [math] , and the light ray transforms of [math] and [math] . Moreover, [math] is an FIO away from the diagonal with a canonical relation given by...

6. #### Propagation of chaos, Wasserstein gradient flows and toric Kähler–Einstein metrics

Berman, Robert J.; Önnheim, Magnus
Motivated by a probabilistic approach to Kähler–Einstein metrics we consider a general nonequilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasiconvex N-particle interaction energy. We show that a deterministic “macroscopic” evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at “negative temperature”, the corresponding limiting evolution equation yields...

7. #### The thin-film equation close to self-similarity

Seis, Christian
We study well-posedness and regularity of the multidimensional thin-film equation with linear mobility in a neighborhood of the self-similar Smyth–Hill solutions. To be more specific, we perform a von Mises change of dependent and independent variables that transforms the thin-film free boundary problem into a parabolic equation on the unit ball. We show that the transformed equation is well-posed and that solutions are smooth and even analytic in time and angular direction. The latter gives the analyticity of level sets of the original equation, and thus, in particular, of the free boundary.

8. #### The shape of low energy configurations of a thin elastic sheet with a single disclination

Olbermann, Heiner
We consider a geometrically fully nonlinear variational model for thin elastic sheets that contain a single disclination. The free elastic energy contains the thickness [math] as a small parameter. We give an improvement of a recently proved energy scaling law, removing the next-to-leading-order terms in the lower bound. Then we prove the convergence of (almost-)minimizers of the free elastic energy towards the shape of a radially symmetric cone, up to Euclidean motions, weakly in the spaces [math] for every [math] , as the thickness [math] is sent to 0.

9. #### Well-posedness and smoothing effect for generalized nonlinear Schrödinger equations

Bienaimé, Pierre-Yves; Boulkhemair, Abdesslam
We improve the result obtained by one of the authors, Bienaimé (2014), and establish the well-posedness of the Cauchy problem for some nonlinear equations of Schrödinger type in the usual Sobolev space [math] for [math] instead of [math] . We also improve the smoothing effect of the solution and obtain the optimal exponent.

10. #### Transference of bilinear restriction estimates to quadratic variation norms and the Dirac–Klein–Gordon system

Candy, Timothy; Herr, Sebastian
Firstly, bilinear Fourier restriction estimates — which are well known for free waves — are extended to adapted spaces of functions of bounded quadratic variation, under quantitative assumptions on the phase functions. This has applications to nonlinear dispersive equations, in particular in the presence of resonances. Secondly, critical global well-posedness and scattering results for massive Dirac–Klein–Gordon systems in dimension three are obtained, in resonant as well as in nonresonant regimes. The results apply to small initial data in scale-invariant Sobolev spaces exhibiting a small amount of angular regularity.

11. #### Nonautonomous maximal \$L^p\$-regularity under fractional Sobolev regularity in time

Fackler, Stephan
We prove nonautonomous maximal [math] -regularity results on UMD spaces, replacing the common Hölder assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on [math] we obtain maximal [math] -regularity for [math] and elliptic operators in divergence form with uniform VMO-modulus in space and [math] -regularity for [math] in time.

12. #### On minimizers of an isoperimetric problem with long-range interactions under a convexity constraint

Goldman, Michael; Novaga, Matteo; Ruffini, Berardo
We study a variational problem modeling the behavior at equilibrium of charged liquid drops under a convexity constraint. After proving the well-posedness of the model, we show [math] -regularity of minimizers for the Coulombic interaction in dimension two. As a by-product we obtain that balls are the unique minimizers for small charge. Eventually, we study the asymptotic behavior of minimizers, as the charge goes to infinity.

13. #### Large sets avoiding patterns

Fraser, Robert; Pramanik, Malabika
We construct subsets of Euclidean space of large Hausdorff dimension and full Minkowski dimension that do not contain nontrivial patterns described by the zero sets of functions. The results are of two types. Given a countable collection of [math] -variate vector-valued functions [math] satisfying a mild regularity condition, we obtain a subset of [math] of Hausdorff dimension [math] that avoids the zeros of [math] for every [math] . We also find a set that simultaneously avoids the zero sets of a family of uncountably many functions sharing the same linearization. In contrast with previous work, our construction allows for nonpolynomial...

14. #### 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...

15. #### 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.

16. #### 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....

17. #### 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]...

18. #### 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.

19. #### 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...

20. #### 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...

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