publications
2024
- Axisymmetric capillary water waves with vorticity and swirl connecting to static unduloid configurationsWith Anna-Mariya Otsetova and Erik WahlénJournal of Differential Equations 411, 604–618
We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. Explicit solutions to such a water wave problem are static configurations where the surface is an unduloid, that is, a periodic surface of revolution with constant mean curvature. We prove that to any such configuration there connects a global continuum of non-static solutions by means of a global implicit function theorem. To prove this, the key is strict monotonicity of a certain function describing the mean curvature of an unduloid and involving complete elliptic integrals. From this point of view, this paper is an interesting interplay between water waves, geometry, and properties of elliptic integrals.
@article{Otsetova_Wahlen_Weber_2024, author = {Otsetova, Anna-Mariya and Wahlén, Erik and Weber, Jörg}, title = {Axisymmetric capillary water waves with vorticity and swirl connecting to static unduloid configurations}, journal = {J. Differential Equations}, fjournal = {Journal of Differential Equations}, volume = {411}, year = {2024}, pages = {604--618}, doi = {10.1016/j.jde.2024.08.005}, } - Controlling a Vlasov–Poisson plasma by a Particle-In-Cell method based on a Monte Carlo frameworkSIAM Journal on Control and Optimization 62 (4), 1977–2011
The Vlasov-Poisson system describes the time evolution of a plasma in the so-called collisionless regime. The investigation of a high-temperature plasma that is influenced by an exterior magnetic field is one of the most significant aspects of thermonuclear fusion research. In this paper, we formulate and analyze a kinetic optimal control problem for the Vlasov-Poisson system where the control is represented by an external magnetic field. The main goal of such optimal control problems is to confine the plasma in a certain region in phase space. We first investigate the optimal control problem in terms of mathematical analysis, i.e., we show the existence of at least one global minimizer and we rigorously derive a first-order necessary optimality condition for local minimizers by the adjoint approach. Then, we build a Monte Carlo framework to solve the state equations as well as the adjoint equations by means of a Particle-In-Cell method, and we apply a nonlinear conjugate gradient method to solve the optimization problem. Eventually, we present numerical experiments that successfully validate our optimization framework.
@article{Bartsch_Knopf_Scheurer_Weber_2024, author = {Bartsch, Jan and Knopf, Patrik and Scheurer, Stefania and Weber, Jörg}, title = {Controlling a Vlasov--Poisson plasma by a Particle-In-Cell method based on a Monte Carlo framework}, journal = {SIAM J. Control Optim.}, fjournal = {SIAM Journal on Control and Optimization}, volume = {62}, year = {2024}, number = {4}, pages = {1977--2011}, doi = {10.1137/23M1563852}, } - Large-amplitude steady gravity water waves with general vorticity and critical layersWith Erik WahlénDuke Mathematical Journal 173 (11), 2197–2258
We consider two-dimensional steady periodic gravity waves on water of finite depth with a prescribed but arbitrary vorticity distribution. The water surface is allowed to be overhanging and no assumptions regarding the absence of stagnation points and critical layers are made. Using conformal mappings and a new reformulation of Bernoulli’s equation, we uncover an equivalent formulation as “identity plus compact,” which is amenable to Rabinowitz’s global bifurcation theorem. This allows us to construct a global connected set of solutions, bifurcating from laminar flows with a flat surface. Moreover, a nodal analysis is carried out for these solutions under a certain spectral assumption involving the vorticity function. Lastly, downstream waves are investigated in more detail.
@article{Wahlen_Weber_2024, author = {Wahlén, Erik and Weber, Jörg}, title = {Large-amplitude steady gravity water waves with general vorticity and critical layers}, journal = {Duke Math. J.}, fjournal = {Duke Mathematical Journal}, volume = {173}, year = {2024}, number = {11}, pages = {2197--2258}, doi = {10.1215/00127094-2023-0054}, }
2023
- Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticityWith Erik WahlénInternational Mathematics Research Notices. IMRN 2023 (20), 17377–17410
We study two-dimensional periodic capillary-gravity water waves propagating at the free surface of water in a flow with arbitrary, prescribed vorticity over a flat bed. Using conformal mappings and a new reformulation of Bernoulli’s equation, the problem is equivalently cast into the form “identity plus compact”, which is amenable to Rabinowitz’s global bifurcation theorem, while no restrictions on the geometry of the surface profile and no assumptions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed.
@article{Wahlen_Weber_2023, author = {Wahlén, Erik and Weber, Jörg}, title = {Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity}, journal = {Int. Math. Res. Not. IMRN}, fjournal = {International Mathematics Research Notices. IMRN}, volume = {2023}, year = {2023}, number = {20}, pages = {17377--17410}, doi = {10.1093/imrn/rnac280}, } - On the amplitude of steady water waves with favorable constant vorticityWith Evgeniy Lokharu and Erik WahlénJournal of Mathematical Fluid Mechanics 25 (3), 58
For two-dimensional steady pure-gravity water waves with a unidirectional flow of constant favorable vorticity, we prove an explicit bound on the amplitude of the wave, which decays to zero as the vorticity tends to infinity. Notably, our result holds true for arbitrary water waves, that is, we do not have to restrict ourselves to periodic or solitary or symmetric waves.
@article{Lokharu_Wahlen_Weber_2023, author = {Lokharu, Evgeniy and Wahlén, Erik and Weber, Jörg}, title = {On the amplitude of steady water waves with favorable constant vorticity}, journal = {J. Math. Fluid Mech.}, fjournal = {Journal of Mathematical Fluid Mechanics}, volume = {25}, year = {2023}, number = {3}, eid = {58}, doi = {10.1007/s00021-023-00796-6}, }
2022
- Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirlWith André H. Erhardt and Erik WahlénStudies in Applied Mathematics 149 (4), 904–942
We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes’ stream function. A change of variables allows us to overcome the generic coordinate-induced singularities and to cast the problem in the form “identity plus compact,” which is amenable to Rabinowitz’s global bifurcation theorem, whereas no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed.
@article{Erhardt_Wahlen_Weber_2022, author = {Erhardt, André H. and Wahlén, Erik and Weber, Jörg}, title = {Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl}, journal = {Stud. Appl. Math.}, fjournal = {Studies in Applied Mathematics}, volume = {149}, year = {2022}, number = {4}, pages = {904--942}, doi = {10.1111/sapm.12525}, } - On the two and one-half dimensional Vlasov–Poisson system with an external magnetic field: global well-posedness and stability of confined steady statesWith Patrik KnopfNonlinear Analysis: Real World Applications 65, 103460
The time evolution of a two-component collisionless plasma is modelled by the Vlasov–Poisson system. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that an external magnetic field is present in order to confine the plasma in a given infinitely long cylinder. After discussing global well-posedness of the corresponding Cauchy problem, we construct stationary solutions whose support stays away from the wall of the confinement device. Then, in the main part of this work we investigate the stability of such steady states, both with respect to perturbations of the initial data, where we employ the energy-Casimir method, and also with respect to perturbations of the external magnetic field.
@article{Knopf_Weber_2022, author = {Knopf, Patrik and Weber, Jörg}, title = {On the two and one-half dimensional Vlasov--Poisson system with an external magnetic field: global well-posedness and stability of confined steady states}, journal = {Nonlinear Anal. Real World Appl.}, fjournal = {Nonlinear Analysis: Real World Applications}, volume = {65}, year = {2022}, eid = {103460}, doi = {10.1016/j.nonrwa.2021.103460}, }
2021
- Weak solutions of the relativistic Vlasov–Maxwell system with external currentsMathematical Methods in the Applied Sciences 44 (6), 4770–4801
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma consists of \(N\) particle species, the particles are located in a bounded container \(Ω⊂\mathbb R^3\), and are subject to boundary conditions on \(∂Ω\). Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We do not impose perfect conductor boundary conditions for the electromagnetic fields but consider the fields as functions on whole space \(\mathbb R^3\) and model objects, that are placed in space, via given matrix-valued functions \(\varepsilon\) (the permittivity) and \(μ\) (the permeability). A weak solution concept is introduced and existence of global-in-time solutions is proved, as well as the redundancy of the divergence part of the Maxwell equations in this weak solution concept.
@article{Weber_2021b, author = {Weber, Jörg}, title = {Weak solutions of the relativistic Vlasov--Maxwell system with external currents}, journal = {Math. Methods Appl. Sci.}, fjournal = {Mathematical Methods in the Applied Sciences}, volume = {44}, year = {2021}, number = {6}, pages = {4770--4801}, doi = {10.1002/mma.7070}, } - Optimal control of the two-dimensional Vlasov-Maxwell systemESAIM. Control, Optimisation and Calculus of Variations 27, S19
The time evolution of a collisionless plasma is modeled by the Vlasov-Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We only consider a two-dimensional version of the problem since existence of global, classical solutions of the full three-dimensional problem is not known. We add external currents to the system, in applications generated by coils, to control the plasma properly. After considering global existence of solutions to this system, differentiability of the control-to-state operator is proved. In applications, on the one hand, we want the shape of the plasma to be close to some desired shape. On the other hand, a cost term penalizing the external currents shall be as small as possible. These two aims lead to minimizing some objective function. We restrict ourselves to only such control currents that are realizable in applications. After that, we prove existence of a minimizer and deduce first order optimality conditions and the adjoint equation.
@article{Weber_2021a, author = {Weber, Jörg}, title = {Optimal control of the two-dimensional Vlasov-Maxwell system}, journal = {ESAIM Control Optim. Calc. Var.}, fjournal = {ESAIM. Control, Optimisation and Calculus of Variations}, volume = {27}, year = {2021}, eid = {S19}, doi = {10.1051/cocv/2020069}, } - A numerical stability analysis for the Einstein–Vlasov systemWith Sebastian Günther, Jacob Körner, Timo Lebeda, Bastian Pötzl, Gerhard Rein and Christopher StraubClassical and Quantum Gravity 38 (3), 035003
We investigate stability issues for steady states of the spherically symmetric Einstein–Vlasov system numerically in Schwarzschild, maximal areal, and Eddington–Finkelstein coordinates. Across all coordinate systems we confirm the conjecture that the first binding energy maximum along a one-parameter family of steady states signals the onset of instability. Beyond this maximum perturbed solutions either collapse to a black hole, form heteroclinic orbits, or eventually fully disperse. Contrary to earlier research, we find that a negative binding energy does not necessarily correspond to fully dispersing solutions. We also comment on the so-called turning point principle from the viewpoint of our numerical results. The physical reliability of the latter is strengthened by obtaining consistent results in the three different coordinate systems and by the systematic use of dynamically accessible perturbations.
@article{Guenther_Koerner_Lebeda_Poetzl_Rein_Straub_Weber_2021, author = {Günther, Sebastian and Körner, Jacob and Lebeda, Timo and Pötzl, Bastian and Rein, Gerhard and Straub, Christopher and Weber, Jörg}, title = {A numerical stability analysis for the Einstein--Vlasov system}, journal = {Classical Quantum Gravity}, fjournal = {Classical and Quantum Gravity}, volume = {38}, year = {2021}, number = {3}, eid = {035003}, doi = {10.1088/1361-6382/abcbdf}, }
2020
- Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinderKinetic and Related Models 13 (6), 1135–1161
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that the plasma is located in an infinitely long cylinder and is influenced by an external magnetic field. We prove existence of stationary solutions and give conditions on the external magnetic field under which the plasma is confined inside the cylinder, i.e., it stays away from the boundary of the cylinder.
@article{Weber_2020b, author = {Weber, Jörg}, title = {Confined steady states of the relativistic Vlasov--Maxwell system in an infinitely long cylinder}, journal = {Kinet. Relat. Models}, fjournal = {Kinetic and Related Models}, volume = {13}, year = {2020}, number = {6}, pages = {1135--1161}, doi = {10.3934/krm.2020040}, } - Optimal control of a Vlasov–Poisson plasma by fixed magnetic field coilsWith Patrik KnopfApplied Mathematics & Optimization 81 (3), 961–988
We consider the Vlasov–Poisson system that is equipped with an external magnetic field to describe the time evolution of the distribution function of a plasma. An optimal control problem where the external magnetic field is the control itself has already been investigated by Knopf (Calc Var 57:134, 2018). However, in real technical applications it will not be possible to choose the control field in such a general fashion as it will be induced by fixed field coils. In this paper we will use the fundamentals that were established by Knopf (Calc Var 57:134, 2018) to analyze an optimal control problem where the magnetic field is a superposition of the fields that are generated by \(N\) fixed magnetic field coils. Thereby, the aim is to control the plasma in such a way that its distribution function matches a desired distribution function at some certain final time \(T\) as closely as possible. This problem will be analyzed with respect to the following topics: existence of a globally optimal solution, necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality and uniqueness of the optimal control under certain conditions.
@article{Knopf_Weber_2020, author = {Knopf, Patrik and Weber, Jörg}, title = {Optimal control of a Vlasov--Poisson plasma by fixed magnetic field coils}, journal = {Appl. Math. Optim.}, fjournal = {Applied Mathematics {\&} Optimization}, volume = {81}, year = {2020}, number = {3}, pages = {961--988}, doi = {10.1007/s00245-018-9526-5}, } - Hot plasma in a container—an optimal control problemSIAM Journal on Mathematical Analysis 52 (3), 2895–2929
The time evolution of a collisionless plasma is modeled by the relativistic Vlasov–Maxwell system which couples the Vlasov equation (the transport equation) with the Maxwell equations of electrodynamics. We consider the case that the plasma is located in a bounded container \(Ω⊂\mathbb R^3\), for example a fusion reactor. Furthermore, there are external currents, typically in the exterior of the container, that may serve as a control of the plasma if adjusted suitably. We model objects, which are placed in space, via given matrix-valued functions \(\varepsilon\) (the permittivity) and \(μ\) (the permeability). A typical aim in fusion plasma physics is to keep the amount of particles hitting \(∂Ω\) as small as possible (since they damage the reactor wall), while the control costs should not be too exhaustive (to ensure efficiency). This leads to a minimizing problem with a PDE constraint. This problem is analyzed in detail. In particular, we prove existence of minimizers and establish an approach to derive first order optimality conditions.
@article{Weber_2020a, author = {Weber, Jörg}, title = {Hot plasma in a container---an optimal control problem}, journal = {SIAM J. Math. Anal.}, fjournal = {SIAM Journal on Mathematical Analysis}, volume = {52}, year = {2020}, number = {3}, pages = {2895--2929}, doi = {10.1137/19M1275061}, }
