We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space ... more We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space Hs(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 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.
We give a natural geometric condition called geodesic compatibility that implies the existence of... more We give a natural geometric condition called geodesic compatibility that implies the existence of integrals in involution of the geodesic flow of a (pseudo)Riemannian metric. We prove that if two metrics satisfy the condition of geodesic compatibility then we can produce a hierarchy of metrics that also satisfy this condition. We apply our results for obtaining an infinite family (hierarchy) of completely integrable flows on the complex projective plane CPn.
We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and in certain... more We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and 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 $\pi/2$ with polynomial growth as $t\to\infty$. We apply these results to nonlinear heat equations on $\R^d$, including global existence in time.
We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on R d wi... more 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| 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.
We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie ... more 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 prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space ... more We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space Hs(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 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.
We give a natural geometric condition called geodesic compatibility that implies the existence of... more We give a natural geometric condition called geodesic compatibility that implies the existence of integrals in involution of the geodesic flow of a (pseudo)Riemannian metric. We prove that if two metrics satisfy the condition of geodesic compatibility then we can produce a hierarchy of metrics that also satisfy this condition. We apply our results for obtaining an infinite family (hierarchy) of completely integrable flows on the complex projective plane CPn.
We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and in certain... more We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and 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 $\pi/2$ with polynomial growth as $t\to\infty$. We apply these results to nonlinear heat equations on $\R^d$, including global existence in time.
We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on R d wi... more 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| 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.
We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie ... more 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.
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