arXiv subjects

Lars Eric Hientzsch

Publications and source records attributed to Lars Eric Hientzsch.

8 recordsLinked to original sources

arXiv2025

Finite energy well-posedness for nonlinear Schrödinger equations with non-vanishing conditions at infinity

Relevant physical phenomena are described by nonlinear Schrödinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the energy space for energy-subcritical nonlinearities, merely satisfying Kato-type assumptions, is proven, providing the analogue of the well-established local $H^1$-theory for solutions vanishing at infinity. The critical nonlinearity will be simply a byproduct of our analysis and the existing literature. Under an assumption that prevents the onset of a Benjamin-Feir type instability, global well-posedness in the energy space is proven for a) non-negative Hamiltonians, b) sign-indefinite Hamiltonians under additional assumptions on the zeros of the nonlinearity, c) generic nonlinearities and small initial data. The cases b) and c) only concern the 3D case

math.AP↗

arXiv2025

Inviscid incompressible limit for capillary fluids with density dependent viscosity

The asymptotic limit of the 2D and 3D Navier-Stokes-Korteweg system for barotropic capillary fluids with density dependent viscosities in the low Mach number and vanishing viscosity regime is established. In the relative energy framework, we prove the convergence of weak solutions of the Navier-Stokes-Korteweg system to the strong solution of the incompressible Euler system. The convergence is obtained through the use of suitable dispersive estimates for an acoustic system altered by the presence of the Korteweg tensor.

math.AP↗

arXiv2025

On the dynamics of leapfrogging vortex rings

The evolution of highly concentrated vorticity around rings in the three-dimensional axisymmetric Euler equations is studied in a regime for which the leapfrogging dynamics predicted by Helmholtz is expected to occur. We provide in this paper the first result deriving this phenomenon for a general class of initial data in the suitable regime. The singular interaction of rings requires significant improvements of weak and strong localization estimates obtained in prior works. Our method is based on the combination of a new variational argument and a recently introduced double iterative procedure.

math.AP↗

arXiv2024

Strong ill-posedness in $L^{\infty}$ of the 2d Boussinesq equations in vorticity form and application to the 3d axisymmetric Euler Equations

We prove the strong ill-posedness in the sense of Hadamard of the two-dimensional Boussinesq equations in $W^{1, \infty}(\mathbb{R}^2)$ without boundary, extending to the case of systems the method that Shikh Khalil \& Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide a large class of initial data with velocity and density of small $W^{1, \infty}(\mathbb{R}^2)$ size, for which the horizontal density gradient has a strong $L^{\infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three-dimensional version of Elgindi's decomposition of the Biot-Savart law, we apply the method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with velocity field uniformly bounded in $W^{1, \infty}(\mathbb{R}^2)$ provides a solution whose swirl component has a strong $W^{1, \infty}(\mathbb{R}^2)$-norm inflation in infinitesimal time, while the potential vorticity remains bounded at least for small times. Finally, the $W^{1,\infty}$-norm inflation of the swirl (and the $L^{\infty}$-norm inflation of the vorticity field) is quantified from below by an explicit lower bound which depends on time, the size of the data and it is valid for small times.

math.AP↗

arXiv2022

Dynamics of several point vortices for the lake equations

The global asymptotic dynamics of point vortices for the lake equations is rigorously derived. Vorticity that is initially sharply concentrated around $N$ distinct vortex centers is proven to remain concentrated for all times. Specifically, we prove weak concentration of the vorticity and in addition strong concentration in the direction of the steepest ascent of the depth function. As a consequence, we obtain the motion law of point vortices following at leading order the level lines of the depth function. The lack of strong localization in the second direction is linked to the vortex filamentation phenomena. The main result allows for any fixed number of vortices and general assumptions on the concentration property of the initial data to be considered. No further properties such as a specific profile or symmetry of the data are required. Vanishing topographies on the boundary are included in our analysis. Our method is inspired by recent results on the evolution of vortex rings in 3D axisymmetric incompressible fluids.

math.AP↗

arXiv2021

On the low Mach number limit for Quantum Navier-Stokes equations

We investigate the low Mach number limit for the 3-D quantum Navier-Stokes system. For general ill-prepared initial data, we prove strong convergence of finite energy weak solutions to weak solutions of the incompressible Navier-Stokes equations. Our approach relies on a quite accurate dispersive analysis for the acoustic part, governed by the well-known Bogoliubov dispersion relation for the elementary excitations of the weakly-interacting Bose gas. Once we have a control of the acoustic dispersion, the a priori bounds provided by the energy and Bresch-Desjardins entropy type estimates lead to the strong convergence. Moreover, for well-prepared data we show that the limit is a Leray weak solution, namely it satisfies the energy inequality. Solutions under consideration in this paper are not smooth enough to allow for the use of relative entropy techniques.

math.AP↗

arXiv2020

Analysis of acoustic oscillations for a class of hydrodynamic systems describing quantum fluids

Hydrodynamic systems for quantum fluids are systems for compressible fluid flows for which quantum effects are macroscopically relevant. We discuss how the presence of the dispersive tensor describing the quantum effects alters the acoustic dispersion at the example of the Quantum Hydrodynamic system (QHD). For the QHD system the dispersion relation is given by the Bogoliubov dispersion relation for weakly interacting Bose gases. We provide refined Strichartz estimates allowing for an accurate control of acoustic oscillations. Applications to the low Mach number limit for the quantum Navier-Stokes equations and the QHD system are discussed. The dispersive analysis generalizes to some Euler- and Navier-Stokes-Korteweg systems for capillary fluids.

math.AP↗

arXiv2020

Global existence of finite energy weak solutions to the Quantum Navier-Stokes equations with non-trivial far-field behavior

We prove global existence of finite energy weak solutions to the quantum Navier-Stokes equations in the whole space with non trivial far-field condition in dimensions d = 2,3. The vacuum regions are included in the weak formulation of the equations. Our method consists in an invading domains approach. More precisely, by using a suitable truncation argument we construct a sequence of approximate solutions. The energy and the BD entropy bounds allow for the passage to the limit in the truncated formulation leading to a finite energy weak solution. Moreover, the result is also valid in the case of compressible Navier-Stokes equations with degenerate viscosity.

math.AP↗