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Peter Topalov

Publications and source records attributed to Peter Topalov.

At least 19 recordsLinked to original sources
arXiv2024

Spatial decay/asymptotics in the Navier-Stokes equation

We discuss the appearance of spatial asymptotic expansions of solutions of the Navier-Stokes equation on $\mathbb{R}^n$. In particular, we prove that the Navier-Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop non-trivial asymptotic terms as $|x|\to\infty$. In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion.

math.AP↗
arXiv2023

On the low regularity phase space of the Benjamin-Ono equation

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 $H^s(\mathbb{T},\mathbb{R})$, $s>-1/2$.

math.AP↗
arXiv2022

Spatially quasi-periodic solutions of the Euler equation

We develop a framework for studying quasi-periodic maps and diffeomorphisms on $\mathbb{R}^n$. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on $\mathbb{R}^n$. 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.

math.AP↗
arXiv2022

Spatial asymptotics and equilibria of heat flow on $\mathbb{R}^d$

We prove that the heat equation on $\mathbb{R}^d$ is well-posed in certain spaces of functions allowing spatial asymptotic expansions as $|x|\to\infty$ 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 $t\to\infty$. 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.

math.AP↗
arXiv2021

On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus

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

math.FA↗
arXiv2021

On smoothing properties and Tao's gauge transform of the Benjamin-Ono equation on the torus

We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space $H^{s}(\mathbb{T},\mathbb{R})$ for any $s\ge 0$. To this end we show that Tao's gauge transform is a high frequency approximation of the nonlinear Fourier transform $Φ$ 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.

math.AP↗
arXiv2021

Perfect fluid flows on $\R^d$ with growth/decay conditions at infinity

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $\R^d$ with initial data in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $|x|^β$ with $β<1/2$. In particular, 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. We identify the evolution space for initial data in the Schwartz class with a certain space of symbols.

math.AP↗
arXiv2020

On the asymptotic behavior of solutions of the 2d Euler equation

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.

math.AP↗
arXiv2019

On the group of A-P diffeomorphisms and its exponential map

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.

math.AP↗
arXiv2019

On the flow map of the Benjamin-Ono equation on the torus

We prove that for any $0 < s < 1/2$, the Benjamin--Ono equation on the torus is globally in time $C^0-$well-posed on the Sobolev space $H^{-s}(\T, \R)$,in the sense that the solution map, which is known to be defined for smooth data, continuously extends to $H^{-s}(\T,\R)$. The solution map does not extend continuously to $H^{-s}(\T, \R)$ with $s > 1/2$. 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 $0\le s<1/2$.

math.AP↗
arXiv2016

Spatial asymptotic expansions in the incompressible Euler equation

In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with initial data in the Schwartz class develop non-trivial spatial asymptotic expansions of the type considered here.

math.AP↗
arXiv2015

Groups of Asymptotic Diffeomorphisms

We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the method of V. Arnold. Specific applications have been obtained for the Camassa-Holm equation and the Euler equations.

math.AP↗
arXiv2015

On the convexity of the KdV Hamiltonian

We prove that the nonlinear part $H^{*}$ of the KdV Hamiltonian $H^{kdv}$, when expressed in action variables $I = (I_{n})_{n\ge 1}$, extends to a real analytic function on the positive quadrant $\ell^2_+(\mathbb N)$ of $\ell^{2}(\mathbb N)$ and is strictly concave near $0$. As a consequence, the differential of $H^{*}$ defines a local diffeomorphism near $0$ of $\ell_{\mathbb C}^{2}(\mathbb N)$.

math.AP↗
arXiv2014

Asymptotics in shallow water waves

In this paper we consider the initial value problem for a family of shallow water equations on the line $\R$ with various asymptotic conditions at infinity. In particular we construct solutions with prescribed asymptotic expansion as $x\to\pm\infty$ and prove their invariance with respect to the solution map.

math.AP↗
arXiv2013

On nonlinear interpolation

In a case study on asymptotics of spectral quantities of Schrödinger operators we show how the Riesz-Thorin theorem on the interpolation of linear operators can be extended to nonlinear maps.

math.FA↗