arXiv subjects

Gabriele Bruell

Publications and source records attributed to Gabriele Bruell.

15 recordsLinked to original sources

arXiv2023

Existence, regularity, and symmetry of periodic traveling waves for Gardner-Ostrovsky type equations

We study the existence, regularity, and symmetry of periodic traveling solutions to a class of Gardner-Ostrovsky type equations, including the classical Gardner-Ostrovsky equation, the (modified) Ostrovsky, and the reduced (modified) Ostrovsky equation. The modified Ostrovsky equation is also known as the short pulse equation. The Gardner-Ostrovsky equation is a model for internal ocean waves of large amplitude. We prove the existence of nontrivial, periodic traveling wave solutions using local bifurcation theory, where the wave speed serves as the bifurcation parameter. Moreover, we give a regularity analysis for periodic traveling solutions in the presence as well as absence of Boussinesq dispersion. We see that the presence of Boussinesq dispersion implies smoothness of periodic traveling wave solutions, while its absence may lead to singularities in the form of peaks or cusps. Eventually, we study the symmetry of periodic traveling solutions by the method of moving planes. A novel feature of the symmetry results in the absence of Boussinesq dispersion is that we do not need to impose a traditional monotonicity condition or a recently developed reflection criterion on the wave profiles to prove the statement on the symmetry of periodic traveling waves.

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arXiv2023

Lump solutions of the fractional Kadomtsev--Petviashvili equation

Of concern are lump solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. As in the classical Kadomtsev--Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case $α>\frac{4}{5}$ by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation nor for the fKP-I when $α\leq \frac{4}{5}$. Furthermore, we show that for any $α>\frac{4}{5}$ lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.

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arXiv2022

Traveling waves for a quasilinear wave equation

We consider a 2+1 dimensional wave equation appearing in the context of polarized waves for the nonlinear Maxwell equations. The equation is quasilinear in the time derivatives and involves two material functions $V$ and $Γ$. We prove the existence of traveling waves which are periodic in the direction of propagation and localized in the direction orthogonal to the propagation direction. Depending on the nature of the nonlinearity coeffcient $Γ$ we distinguish between two cases: (a) $Γ\in L^\infty$ being regular and (b) $Γ=γδ_0$ being a multiple of the delta potential at zero. For both cases we use bifuraction theory to prove the existence of nontrivial small-amplitude solutions. One can regard our results as a persistence result which shows that guided modes known for linear wave-guide geometries survive in the presence of a nonlinear constitutive law. Our main theorems are derived under a set of conditions on the linear wave operator. They are subsidised by explicit examples for the coefficients $V$ in front of the (linear) second time derivative for which our results hold.

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arXiv2022

Traveling waves and transverse instability for the fractional Kadomtsev-Petviashvili equation

Of concern are traveling wave solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. The existence of periodically modulated solitary wave solutions is proved by dimension breaking bifurcation. Moreover, the line solitary wave solutions and their transverse (in)stability are discussed. Analogous to the classical Kadmomtsev--Petviashvili (KP) equation, the fKP equation comes in two versions: fKP-I and fKP-II. We show that the line solitary waves of fKP-I equation are transversely linearly instable. We also perform numerical experiments to observe the (in)stability dynamics of line solitary waves for both fKP-I and fKP-II equations.

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arXiv2021

Non-Newtonian two-phase thin-film problem: Local existence, uniqueness, and stability

We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the two-phase Navier--Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.

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arXiv2021

Symmetry of periodic traveling waves for nonlocal dispersive equations

Of concern is the a priori symmetry of traveling wave solutions for a general class of nonlocal dispersive equations \[ u_t +(u^2 +Lu)_x =0, \] where $L$ is a Fourier multiplier operator with symbol $m$. Our analysis includes both homogeneous and inhomogeneous symbols. We characterize a class of symbols $m$ guaranteeing that periodic traveling wave solutions are symmetric under a mild assumption on the wave profile. Particularly, instead of considering waves with a unique crest and trough per period or a monotone structure near troughs as classically imposed in the water wave problem, we formulate a reflection criterion, which allows to affirm the symmetry of periodic traveling waves. The reflection criterion weakens the assumption of monotonicity between trough and crest and enables to treat a priori solutions with multiple crests of different sizes per period. Moreover, our result not only applies to smooth solutions, but also to traveling waves with a non-smooth structure such as peaks or cusps at a crest. The proof relies on a so-called touching lemma, which is related to a strong maximum principle for elliptic operators, and a weak form of the celebrated method of moving planes.

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arXiv2019

Well-posedness and stability for a mixed order system arising in thin film equations with surfactant

The objective of the present work is to provide a well-posedness result for a capillary driven thin film equation with insoluble surfactant. The resulting parabolic system of evolution equations is not only strongly coupled and degenerated, but also of mixed orders. To the best of our knowledge the only well-posedness result for a capillary driven thin film with surfactant is provided in [4] by the same author, where a severe smallness condition on the surfactant concentration is assumed to prove the result. Thus, in spite of an intensive analytical study of thin film equations with surfactant during the last decade, a proper well-posedness result is still missing in the literature. It is the aim of the present paper to fill this gap. Furthermore, we apply a recently established result on asymptotic stability in interpolation spaces [15] to prove that the flat equilibrium of our system is asymptotically stable.

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arXiv2019

On a thin film model with insoluble surfactant

This paper studies the existence and asymptotic behavior of global weak solutions for a thin film equation with insoluble surfactant under the influence of gravitational, capillary and van der Waals forces. We prove the existence of global weak solutions for \emph{medium sized} initial data in \emph{large function spaces}. Moreover, exponential decay towards the flat equilibrium state is established, where an estimate on the decay rate can be computed explicitly.

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arXiv2018

Waves of maximal height for a class of nonlocal equations with homogeneous symbols

We discuss the existence and regularity of periodic traveling-wave solutions of a class of nonlocal equations with homogeneous symbol of order $-r$, where $r>1$. Based on the properties of the nonlocal convolution operator, we apply analytic bifurcation theory and show that a highest, peaked, periodic traveling-wave solution is reached as the limiting case at the end of the main bifurcation curve. The regularity of the highest wave is proved to be exactly Lipschitz. As an application of our analysis, we reformulate the steady reduced Ostrovsky equation in a nonlocal form in terms of a Fourier multiplier operator with symbol $m(k)=k^{-2}$. Thereby we recover its unique highest $2π$-periodic, peaked traveling-wave solution, having the property of being exactly Lipschitz at the crest.

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arXiv2018

On a singular limit for stratified compressible fluids

We consider a singular limit problem for the complete compressible Euler system in the low Mach and strong stratification regime. We identify the limit problem - the anelastic Euler system - in the case of well prepared initial data. The result holds in the large class of the dissipative measure-valued solutions of the primitive system. Applications are discussed to the driven shallow water equations.

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arXiv2018

On the thin film Muskat and the thin film Stokes equations

The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase thin film flows. In particular, we consider the two-phase thin film Muskat problem and the two-phase thin film approximation of the Stokes flow under the influence of both, capillary and gravitational forces. The existence of global weak solutions for medium size initial data in large function spaces is proved. Moreover, exponential decay results towards the equilibrium state are established, where the decay rate can be estimated by explicit constants depending on the physical parameters of the system. Eventually, it is shown that if the initial datum satisfies additional (low order) Sobolev regularity, we can propagate Sobolev regularity for the corresponding solution. The proofs are based on a priori energy estimates in Wiener and Sobolev spaces.

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arXiv2017

Symmetric solutions of evolutionary partial differential equations

We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms of three principles, based on the structure of the equations. The first principle covers equations that allow for steady solutions and shows that any spatially symmetric solution is in fact steady with a speed determined by the motion of the axis of symmetry at the initial time. The second principle includes equations that admit breathers and steady waves, and therefore is less strong: it holds that the axes of symmetry are constant in time. The last principle is a mixed case, when the equation contains terms of the kind from both earlier principles, and there may be different outcomes; for a class of such equations one obtains that a spatially symmetric solution must be constant in both time and space. We list and give examples of more than 30 well-known equations and systems in one and several dimensions satisfying these principles; corresponding results for weak formulations of these equations may be attained using the same techniques. Our investigation is a generalisation of a local and one-dimensional version of the first principle from [E., Holden, and Raynaud, 2009] to nonlocal equations, systems and higher dimensions, as well as a study of the standing and mixed cases.

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arXiv2017

Symmetry and decay of traveling wave solutions to the Whitham equation

This paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model. It is shown that all supercritical solitary wave solutions are symmetric and monotone on either side of the crest. The proof is based on a priori decay estimates and the method of moving planes. Furthermore, a close relation between symmetric and traveling wave solutions is established.

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arXiv2016

Weak solutions to a two-phase thin film model with insoluble surfactant driven by capillary effects

Of concern is the study of a system of three equations describing the motion of a viscous complete wetting two-phase thin film endowed with a layer of insoluble surfactant on the surface of the upper fluid under the effects of capillary forces. The governing equations for the film heights of the two-phase flow are degenerate, parabolic and strongly coupled fourth-order equations, which are additionally coupled to a second-order parabolic transport equation for the surfactant concentration. A result on the existence of non-negative global weak solutions is presented.

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arXiv2016

Modeling and Analysis of a Two-Phase Thin Film Model with Insoluble Surfactant

In this paper we consider a two-phase thin film consisting of two immiscible viscous fluids endowed with a layer of insoluble surfactant on the surface of the upper fluid. The governing equations for the two film heights and the surfactant concentration are derived using a lubrication approximation. Taking gravitational forces into account but neglecting capillary effects, the resulting system of evolution equations is parabolic, strongly coupled, of second order and degenerated in the equations for the two film heights. Incorporating on the contrary capillary forces and neglecting the effects of gravitation, the system of evolution equations is parabolic, degenerated and of fourth order for the film heights, strongly coupled to a second order transport equation for the surfactant concentration. Local well-posedness and asymptotic stability are shown for both systems.

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