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Björn Gebhard

Publications and source records attributed to Björn Gebhard.

14 recordsLinked to original sources
arXiv2026

On Leonardo da Vinci's paradox

Buoyancy points straight upward, yet in the absence of external forces, a rising air bubble may trace a spiral through the water. This phenomenon was already observed by Leonardo da Vinci. We show that the gravity-free, three-dimensional two-phase incompressible Euler equations with surface tension already admit such a motion. For every nonnegative inner-to-outer density ratio and every sufficiently small positive axial Weber number, we construct smooth near-spherical bubbles and drops with compact interfaces and phasewise irrotational flow. They are stationary in a frame translating along and rotating about a fixed axis, while their centroids lie off that axis and therefore trace genuine circular helices in laboratory coordinates. These helical relative equilibria form a symmetry-breaking branch that issues from the axisymmetric family of rectilinearly translating solutions and is generated by the spherical harmonic mode of degree two and azimuthal number one. The mathematical construction exploits an overdetermined free-boundary formulation, its variational structure, and an $\mathrm{SO}(2)$-equivariant Lyapunov-Schmidt reduction. Particular difficulties in the form of resonances arise at a discrete set of density ratios, including the frequently studied vacuum case with vanishing inner density. Complementary to our existence result, in the absence of surface tension and if the inner density is not larger than the outer density we prove that the model admits neither nontrivial translating relative equilibria nor relative equilibria with helical centroid trajectories. As a byproduct, we show that every regular finite-energy stationary vortex sheet of spherical topology for the homogeneous three-dimensional Euler equations, with irrotational flow on both sides, is trivial.

math.AP↗
arXiv2025

The Rayleigh-Taylor instability with local energy dissipation

We consider the inhomogeneous incompressible Euler equations including their local energy inequality as a differential inclusion. Providing a corresponding convex integration theorem and constructing subsolutions, we show the existence of locally dissipative Euler flows emanating from the horizontally flat Rayleigh-Taylor configuration and having a mixing zone which grows quadratically in time. For the Rayleigh-Taylor instability these are the first turbulently mixing solutions known to respect local energy dissipation, and outside the range of Atwood numbers considered in arXiv:2002.08843, the first weakly admissible solutions in general. In the coarse grained picture the existence relies on one-dimensional subsolutions described by a family of hyperbolic conservation laws, among which one can find the optimal background profile appearing in the scale invariant bounds from arXiv:2303.01889, and as we show, the optimal conservation law with respect to maximization of the total energy dissipation.

math.AP↗
arXiv2024

Entropy solutions to macroscopic IPM

We investigate maximal potential energy dissipation as a selection criterion for subsolutions (coarse grained solutions) in the setting of the unstable Muskat problem. We show that both, imposing this criterion on the level of convex integration subsolutions, and the strategy of Otto based on a relaxation via minimizing movements, lead to the same nonlocal conservation law. Our main result shows that this equation admits an entropy solution for unstable initial data with an analytic interface.

math.AP↗
arXiv2024

On the energy-constrained optimal mixing problem for one-dimensional initial configurations

We consider the problem of mixing a passive scalar in a periodic box by incompressible vector fields subject to a fixed energy constraint. In that setting a lower bound for the time in which perfect mixing can be achieved has been given by Lin, Thiffeault, Doering \cite{Lin_Thiffeault_Doering_2011}. While examples by Depauw \cite{Depauw} and Lunasin et al. \cite{Lunasin_etal_2012} show that perfect mixing in finite time is indeed possible, the question regarding the sharpness of the lower bound from \cite{Lin_Thiffeault_Doering_2011} remained open. In the present article we give a negative answer for the special class of initial configurations depending only on one spatial coordinate. The new lower bound holds true for distributional solutions satisfying only the uniform energy constraint for the velocity field and a weak compatibility condition for the passive scalar coming from the transport equation. In that weak setting we also provide an example for which the new bound is sharp. As a new ingredient in the investigation of optimal mixing we utilize the convex hull inequalities of the transport equation with constraints when seen as a differential inclusion.

math.AP↗
arXiv2022

On a degenerate elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions

We address a degenerate elliptic variational problem arising in the application of the least action principle to averaged solutions of the inhomogeneous Euler equations in Boussinesq approximation emanating from the horizontally flat Rayleigh-Taylor configuration. We give a detailed derivation of the functional starting from the differential inclusion associated with the Euler equations, i.e. the notion of an averaged solution is the one of a subsolution in the context of convex integration, and illustrate how it is linked to the generalized least action principle introduced by Brenier in \cite{Brenier89,Brenier18}. Concerning the investigation of the functional itself, we use a regular approximation in order to show the existence of a minimzer enjoying partial regularity, as well as other properties important for the construction of actual Euler solutions induced by the minimizer. Furthermore, we discuss to what extent such an application of the least action principle to subsolutions can serve as a selection criterion.

math.AP↗
arXiv2021

On bounded two-dimensional globally dissipative Euler flows

We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered without local energy (im)balance. Concerning bounded solutions we provide a sufficient criterion for a globally dissipative subsolution to induce infinitely many globally dissipative solutions having the same initial data, pressure and dissipation measure as the subsolution. The criterion can easily be verified in the case of a flat vortex sheet giving rise to the Kelvin-Helmholtz instability. As another application we show that there exists initial data, for which associated globally dissipative solutions realize every dissipation measure from an open set in $\mathcal{C}^0(\mathbb{T}^2\times[0,T])$. In fact the set of such initial data is dense in the space of solenoidal $L^2(\mathbb{T}^2;\mathbb{R}^2)$ vector fields.

math.AP↗
arXiv2020

A general way to confined stationary Vlasov-Poisson plasma configurations

We address the existence of stationary solutions of the Vlasov-Poisson system on a domain $Ω\subset\mathbb{R}^3$ describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of $Ω$. In a first part we provide such an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field, the sharpness of the confinement and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.

math.AP↗
arXiv2020

Relaxation of the Boussinesq system and applications to the Rayleigh-Taylor instability

We consider the evolution of two incompressible fluids with homogeneous densities $ρ_-<ρ_+$ subject to gravity described by the inviscid Boussinesq equations and provide the explicit relaxation of the associated differential inclusion. The existence of a subsolution to the relaxation allows one to conclude the existence of turbulently mixing solutions to the original Boussinesq system. As a specific application we investigate subsolutions emanating from the classical Rayleigh-Taylor initial configuration where the two fluids are separated by a horizontal interface with the heavier fluid being on top of the lighter. It turns out that among all self-similar subsolutions the criterion of maximal initial energy dissipation selects a linear density profile and a quadratic growth of the mixing zone. The subsolution selected this way can be extended in an admissible way to exist for all times. We provide two possible extensions with different long-time limits. The first one corresponds to a total mixture of the two fluids, the second corresponds to a full separation with the lighter fluid on top of the heavier. There is no motion in either of the limit states.

math.AP↗
arXiv2020

A new approach to the Rayleigh-Taylor instability

In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one - the classical configuration giving rise to the Rayleigh-Taylor instability. We construct specific examples when the Atwood number is in the ultra high range, for which the zone in which the mixing occurs grows quadratically in time.

math.AP↗
arXiv2018

Stability of periodic solutions of the N-vortex problem in general domains

We investigate stability properties of a type of periodic solutions of the $N$-vortex problem on general domains $Ω\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point $a_0\inΩ$ of the Robin function associated to the Dirichlet Laplacian of $Ω$. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point $a_0$. If $a_0$ is a saddle, they are unstable. Otherwise they are stable in a certain linear sense. The proof uses a criterion for the bifurcation of multiple eigenvalues, which is applied to suitable Poincaré sections. Beyond linear stability, Herman's last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of $N=2$ vortices.

math.DS↗
arXiv2017

Periodic solutions for the N-vortex problem via a superposition principle

We examine the $N$-vortex problem on general domains $Ω\subset\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form $$ Γ_k\dot{z}_k=J\nabla_{z_k}H(z_1,\ldots,z_N),\quad k=1,\ldots,N, $$ where $Γ_k\in\mathbb{R}\setminus\{0\}$ is the strength of the $k$th vortex at position $z_k(t)\inΩ$, $J\in\mathbb{R}^{2\times 2}$ is the standard symplectic matrix and $$ H(z_1,\ldots,z_N)=-\frac{1}{2π}\sum_{\underset{k\neq j}{k,j=1}}^NΓ_jΓ_k\log|z_k-z_j|-\sum_{k,j=1}^NΓ_jΓ_k g(z_k,z_j) $$ with some regular and symmetric, but in general not explicitely known function $g:Ω\timesΩ\rightarrow \mathbb{R}$. The investigation relies on the idea to superpose a stationary solution of a system of less than $N$ vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple $T$-periodic solutions are shown to exist for every $T>0$ small enough. The crucial condition holds in generic bounded domains and is explicitely verified for an example in the unit disc $Ω=B_1(0)$. In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.

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arXiv2016

Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary

The paper deals with the existence of nonstationary collision-free periodic solutions of singular first order Hamiltonian systems of $N$-vortex type in a domain $Ω\subset\mathbb{C}$. These are solutions $z(t)=(z_1(t),\dots,z_N(t))$ of \[ \dot{z}_j(t)=-i\nabla_{z_j} H_Ω\big(z(t)\big),\quad j=1,\dots,N, \tag{HS} \] where the Hamiltonian $H_Ω$ has the form \[ H_Ω(z_1,\dots,z_N) = -\sum_{{j,k=1}\over{j\ne k}}^N \frac{1}{2π}\log|z_j-z_k| -\sum_{j,k=1}^N g(z_j,z_k). \] The function $g:Ω\timesΩ\to\mathbb{R}$ is required to be of class $C^3$ and symmetric, the regular part of a hydrodynamic Green function being our model. The Hamiltonian is unbounded from above and below, and the associated action integral is not defined on an open subset of the space of periodic $H^{1/2}$ functions. Given a closed connected component $Γ\subset\partialΩ$ of class $C^3$ we are interested in periodic solutions of (HS) near $Γ$. We present quite general conditions on the behavior of $g$ near $Γ$ which imply that there exists a family of periodic solutions $z^{(r)}(t)$, $0<r<\overline{r}$, with arbitrarily small minimal period $T_r\to0$ as $r\to0$, and such that the "point vortices" $z_j^{(r)}(t)$ approach $Γ$ as $r\to0$. The solutions are choreographies, i.e.\ $z_j^{(r)}(t)$ moves on the same trajectory as $z_1^{(r)}(t)$ with a phase shift. We can also relate the speed of each vortex with the curvature of $Γ$.

math.DS↗
arXiv2016

Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

The paper deals with singular first order Hamiltonian systems of the form \[ Γ_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in Ω\subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in\mathbb{R}^{2\times2}$ defines the standard symplectic structure in $\mathbb{R}^2$, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2π} \sum_{j\neq k=1}^N Γ_j Γ_k \log|z_j-z_k| - F(z). \] This is defined on the configuration space $\{(z_1,\ldots,z_N)\in Ω^{2N}:z_j\neq z_k\text{ for }j\neq k\}$ of $N$ different points in the domain $Ω\subset\mathbb{R}^2$. The function $F:Ω^N\to\mathbb{R}$ may have additional singularities near the boundary of $Ω^N$. We prove the existence of a global continuum of periodic solutions $z(t)=(z_1(t),\dots,z_N(t))\inΩ^N$ that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium $Z(t)\in\mathbb{R}^{2N}$ of the $N$-vortex problem in the whole plane (where $F=0$). Examples for $Z$ include Thomson's vortex configurations, or equilateral triangle solutions. The domain $Ω$ need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of $2π$-periodic $H^{1/2}$ functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for $S^1$-equivariant gradient maps that we adapt to this class of potential operators.

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