Related papers: Quasi-periodic incompressible Euler flows in 3D
We construct time almost-periodic solutions (global in time) with finite regularity to the incompressible Euler equations on the torus $\T^d$, with $d=3$ and $d\in\N$ even.
Building on the work of Crouseilles and Faou on the 2D case, we construct $C^\infty$ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions…
Recently, two of these authors construct dissipative continuous (weak) solutions to the incompressible Euler equations on the three-dimensional torus $\mathbb T^3$. The building blocks in their proof are Beltrami flows, which are inherently…
In this paper we consider the incompressible 2D Euler equation in an annular domain with non-penetration boundary condition. In this setting, we prove the existence of a family of non-trivially smooth time-periodic solutions at an…
A novel derivation of non-stationary solutions of 3D Euler equations for incompressible inviscid flow is considered here. Such a solution is the product of 2 separated parts: - one consisting of the spatial component and the other being…
Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we construct global weak solutions to the three-dimensional incompressible Euler equations which are zero outside of a finite time interval and have velocity in the…
Recently, De Lellis and Sz\'ekelyhidi constructed H\"older continuous, dissipative (weak) solutions to the incompressible Euler equations in the torus $\mathbb T^3$. The construction consists in adding fast oscillations to the trivial…
We construct time quasi-periodic solutions to nonlinear wave equations on the torus in arbitrary dimensions. All previously known results (in the case of zero or a multiplicative potential) seem to be limited to the circle. This generalizes…
We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy.
We construct time quasi-periodic solutions to the energy supercritical nonlinear Schr\"odinger equations on the torus in arbitrary dimensions. This introduces a new approach, which could have general applicability.
We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier-Stokes equation on the $d$-dimensional torus $\T^d$, with a small, quasi-periodic in time external force. We also show…
We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or H\"older continuous for any exponent $\theta<\frac{1}{16}$. Using the techniques introduced in \cite{DS12}…
We consider the smooth, compactly supported solutions of the steady 3D Euler equations of incompressible fluids constructed by Gavrilov in 2019, and we study the corresponding fluid particle dynamics. This is an ode analysis, which…
The main result of this research Monograph is the existence of small amplitude time quasi-periodic solutions for autonomous nonlinear wave equations $$ u_{tt} - \Delta u + V(x) u + g(x, u) = 0 \, , \quad x \in T^d \, , \quad g (x,u) = a(x)…
We demonstrate the existence of smooth three-dimensional vector fields where the cross product between the vector field and its curl is balanced by the gradient of a smooth function, with toroidal level sets that are not invariant under…
We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time…
In this paper, we study the control system associated with the incompressible 3D Euler system. We show that the velocity field and pressure of the fluid are exactly controllable in projections by the same finite-dimensional control.…
Under the action of a time-periodic external forces we prove the existence of at least one time-periodic weak solution for the interaction between a three-dimensional incompressible fluid, governed by the Navier- Stokes equation and a two…
A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to…
In this paper, we prove the existence of quasi-periodic small-amplitude solutions for quasi-linear Hamiltonian perturbation of the fifth-order KdV equation on the torus in presence of a quasi-periodic forcing.