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Methods Of Modern Mathematical Physics

Stochastic equations in infinite dimensions

We prove the l 2 Decoupling Conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which implies the full range of expected L p Strichartz estimates for both the rational and (up to N " losses) the irrational torus. Another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere. Various applications to Additive Combinatorics, Incidence Geometry and Number Theory are also discussed. Our argument relies on the interplay between linear and multilinear restriction theory.

http://annals.math.princeton.edu/wp-content/uploads/Bourgain_Demeter.pdf

The proof of the l 2 Decoupling Conjecture

In this paper we study the existence of global solutions to the Cauchy problem of nonlinear SchrSdinger equation by establishing time weight function spaces and using the contraction mapping principle.

Global solutions of nonlinear schrodinger equations

We consider the Cauchy problem for the one-dimensional periodic cubic non- linear Schrodinger equation (NLS) with initial data below L 2 . In particular, we exhibit nonlinear smoothing when the initial data are randomized. Then, we prove local well- posedness of NLS almost surely for the initial data in the support of the canonical Gaussian measures on H s (T) for each s > − 1 , and global well-posedness for each s > − 1 12 .

Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^{2}(\mathbb{T})$

We consider the Cauchy problem of the cubic nonlinear Schrodinger equation (NLS) : i∂tu+Δu = ±|u|2u on Rd, d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity scrit = d−2 2 . More precisely, given a function on Rd, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scalingcritical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove ‘conditional’ almost sure global well-posedness for d = 4 in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when d = 4, we show that conditional almost sure global wellposedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

/pdf/on-the-probabilistic-cauchy-theory-of-the-cubic-nonlinear-1qw005m080.pdf

On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3

In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schr\"odinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier-Lebesgue space ${\mathcal F}L^{s,r}(\T)$ with $s \ge \frac{1}{2}$, $2 0$. We also show the invariance of this measure.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=14&iss=4&rank=9

Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

We study the two-dimensional stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing. In particular, we introduce a time-dependent renor- malization and prove that SNLW is pathwise locally well-posed. As an application of the local well-posedness argument, we also establish a weak universality result for the renormalized SNLW.

Renormalization of the two-dimensional stochastic nonlinear wave equations

We consider the cubic–quintic nonlinear Schrodinger equation: 

$$i\partial_t u = -\Delta u - |u|^2u + |u|^4u.$$

In the first part of the paper, we analyze the one-parameter family of ground state solitons associated to this equation with particular attention to the shape of the associated mass/energy curve. Additionally, we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one-parameter family of inequalities of Gagliardo–Nirenberg type. Building on this work, in the latter part of the paper we prove that scattering holds for solutions belonging to the region ${{\mathcal{R}}}$ of the mass/energy plane where the virial is positive. We show that this region is partially bounded by solitons also by rescalings of solitons (which are not soliton solutions in their own right). The discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial-based methodology.

/pdf/solitons-and-scattering-for-the-cubic-quintic-nonlinear-15lgrb2zs8.pdf

Tadahiro Oh

Papers

Almost sure well-posedness of the cubic nonlinear Schrödinger equation below $L^{2}(\mathbb{T})$

On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3

Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS

Renormalization of the two-dimensional stochastic nonlinear wave equations

Solitons and Scattering for the Cubic–Quintic Nonlinear Schrödinger Equation on $${\mathbb{R}^3}$$ R 3