In this paper, we prove that the Benjamin–Ono equation is globally in time C^{0} -well-posed in the Hilbert space H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R}) of periodic distributions in H^{-1/2}(\mathbb{T},\mathbb{R}) with \sqrt{\log} -weights. The space H^{-1/2,\sqrt{\log}}(\mathbb{T},\mathbb{R}) can thus be considered as a maximal low regularity phase space for the Benjamin–Ono equation corresponding to the scale H^{s}(\mathbb{T},\mathbb{R}) , s>-1/2 .

We prove that the nonlinear Fourier transform of the Benjamin-Ono equation on 𝕋 , also referred to as Birkhoff map, is a real analytic diffeomorphism from the scale of Sobolev spaces H^s_0(𝕋, ℝ) , s > -1/2 , to the scale of weighted ℓ ^2- sequence spaces, 𝔥^s +1/2_r,0(ℕ, ℂ) , s >-1/2 . As an application we show that for any -1/2

We prove that the Navier-Stokes equation is well-posed in function spaces on ℝ^d, d≥ 2, that contain vector fields of order O(|x|^κ) as |x|→∞ with κ<1/2. The corresponding solutions depend continuously on the viscosity parameter ν≥ 0 and converge to the solutions of the Euler equation as ν→ 0+. Our proof is based on the properties of the conjugated heat flow on weighted Sobolev spaces and on a new variant of the Lie-Trotter product formula for nonlinear semigroups.

We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space H-s(T, R) for any s >= 0. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourier transform for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.

We prove that the Navier-Stokes equation for a viscous incompressible fluid in R-d is locally well-posed in spaces of functions allowing spatial asymptotic expansions with log terms as |x|->infinity of any a priori given order. The solution depends analytically on the initial data and time so that for any 0<v<pi/2 it can be holomorphically extended in time to a conic sector in C with angle 2v at zero. We discuss the approximation of solutions by their asymptotic parts.

In this paper we show that the billiard ball map of the Liouville billiard tables of classical type on the ellipsoid is non-degenerate at the elliptic fixed point. As a corollary we obtain a spectral rigidity result.

We prove that the Benjamin–Ono equation on the torus is globally in time well-posed in the Sobolev space H^s(𝕋,ℝ) for any s > - 1/2 and ill-posed for s ≤ - 1/2. Hence the critical Sobolev exponent s_c=-1/2 of the Benjamin–Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in H^s(𝕋,ℝ) for any s > - 1/2. Novel conservation laws and a nonlinear Fourier transform on H^s(𝕋,ℝ) with s > - 1/2 are key ingredients into the proofs of these results.

We prove that the heat equation on Rd is well-posed in certain spaces of functions allowing spatial asymptotic expansions as |x|→∞ of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle π/2 with polynomial growth as t→∞. Generically, a large class of nonlinear heat flows have equilibrium solutions with spatial asymptotics of the considered type. We provide a simple nonlinear model that features global in time existence with such asymptotics at spatial infinity.

We develop a framework for studying quasi-periodic maps and diffeomorphisms on ℝ^n. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on ℝ^n. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved.

In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H-0(s) (T, R) for any s > -1/2 where H-0(s) (T, R) denotes the subspace of the Sobolev space H-s(T, R) of elements with mean 0. As an application we show that for any -1/2 < s < 0, the flow map of the Benjamin-Ono equation S-0(t) : H-0(s)(T, R) -> H-0(s) (T, R) is nowhere locally uniformly continuous in a neighborhood of zero in H-0(s)(T, R). (c) 2021 Published by Elsevier Ltd.

In this paper, we survey our recent results on the Benjamin-Ono equation on the torus. As an application of the methods developed we construct large families of periodic or quasiperiodic solutions, which are not C∞-smooth.

We prove an infinite-dimensional version of the Arnold-Liouville theorem for integrable nonlinear PDEs. In a case study we consider the focusing NLS equation with periodic boundary conditions.

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on ℝ^d with initial velocity in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as |x|^β with β <1/2 . Moreover, for initial velocity with sufficient spatial decay, we show that the solution of the Euler equation generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data. For initial data in the Schwartz space, we identify the evolution space of the fluid velocity with a certain space of symbols.

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $$\mathbb {R}^d$$ with initial velocity in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $$|x|^\beta $$ with $$\beta <1/2$$ . Moreover, for initial velocity with sufficient spatial decay, we show that the solution of the Euler equation generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data. For initial data in the Schwartz space, we identify the evolution space of the fluid velocity with a certain space of symbols.

We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.

We investigate the spectrum of the Lax operator Lu of the Benjamin-Ono equation on the torus for complex valued potentials u in the Sobolev space H−s(T,C), 0≤s<1/2, with small imaginary part and prove analytic properties of the moment map, defined in terms of spectral data of Lu.

For the focusing NLS and the focusing mKdV equation on the circle we present an infinite dimensional version of the Arnold-Liouville theorem.

We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of t, do not involve (spacial) logarithmic terms, and develop even when the initial data has fast decay at infinity. We discuss the evolution in time of the asymptotic terms and their approximation properties.

We prove that the Benjamin--Ono equation on the torus is globally in time well-posed in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s > - 1/2$ and ill-posed for $s \le - 1/2$. Hence the critical Sobolev exponent $s_c=-1/2$ of the Benjamin--Ono equation is the threshold for well-posedness on the torus. The obtained solutions are almost periodic in time. Furthermore, we prove that the traveling wave solutions of the Benjamin-Ono equation on the torus are orbitally stable in $H^{s}(\mathbb{T},\mathbb{R})$ for any $ s > - 1/2$. Novel conservation laws and a nonlinear Fourier transform on $H^{s}(\mathbb{T},\mathbb{R})$ with $s > - 1/2$ are key ingredients into the proofs of these results.