Tag Archives: Analysis of PDEs

Spectrum of the linearized Vlasov–Poisson equation around steady states from galactic dynamics [CL]

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http://arxiv.org/abs/2305.05749


We study the linearized Vlasov-Poisson equation in the gravitational case around steady states that are decreasing and continuous functions of the energy. We identify the absolutely continuous spectrum and give criteria for the existence of oscillating modes and estimate their number. Our method allows us to take into account an attractive external potential.

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M. Moreno, P. Rioseco and H. Bosch
Thu, 11 May 23
5/55

Comments: 17 pages, 2 figures

Turning point principle for stability of viscous gaseous stars [CL]

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http://arxiv.org/abs/2301.07328


We consider stability of non-rotating viscous gaseous stars modeled by the Navier-Stokes-Poisson system. Under general assumptions on the equations of states, we proved that the number of unstable modes for the linearized Navier-Stokes-Poisson system equals that of the linearized Euler-Poisson system modeling inviscid gaseous stars. In particular, the turning point principle holds true for non-rotating stars with or without viscosity. That is, the transition of stability only occurs at the extrema of the total mass and the number of unstable modes is determined by the mass-radius curve. For the proof, we establish an infinite dimensional Kelvin-Tait-Chetaev theorem for a class of linear second order PDEs with dissipation. Moreover, we prove that linear stability implies nonlinear asymptotic stability and linear instability implies nonlinear instability for Navier-Stokes-Poisson system.

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M. Cheng, Z. Lin and Y. Wang
Thu, 19 Jan 23
54/100

Comments: N/A

Stability of rotating gaseous stars [CL]

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http://arxiv.org/abs/2209.00171


We consider stability of rotating gaseous stars modeled by the Euler-Poisson system with general equation of states. When the angular velocity of the star is Rayleigh stable, we proved a sharp stability criterion for axi-symmetric perturbations. We also obtained estimates for the number of unstable modes and exponential trichotomy for the linearized Euler-Poisson system. By using this stability criterion, we proved that for a family of slowly rotating stars parameterized by the center density with fixed angular velocity profile, the turning point principle is not true. That is, unlike the case of non-rotating stars, the change of stability of the rotating stars does not occur at extrema points of the total mass. By contrast, we proved that the turning point principle is true for the family of slowly rotating stars with fixed angular momentum distribution. When the angular velocity is Rayleigh unstable, we proved linear instability of rotating stars. Moreover, we gave a complete description of the spectra and sharp growth estimates for the linearized Euler-Poisson equation.

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Z. Lin and Y. Wang
Fri, 2 Sep 22
26/62

Comments: N/A

Instabilities Appearing in Effective Field theories: When and How? [CL]

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http://arxiv.org/abs/2205.01055


Nonlinear partial differential equations appear in many domains of physics, and we study here a typical equation which one finds in effective field theories (EFT) originated from cosmological studies. In particular, we are interested in the equation $\partial_t^2 u(x,t) = \alpha (\partial_x u(x,t))^2 +\beta \partial_x^2 u(x,t)$ in $1+1$ dimensions. It has been known for quite some time that solutions to this equation diverge in finite time, when $\alpha >0$. We study the detailed nature of this divergence as a function of the parameters $\alpha>0 $ and $\beta\ge0$. The divergence does not disappear even when $\beta $ is very large contrary to what one might believe. But it will take longer to appear as $\beta$ increases when $\alpha$ is fixed. We note that there are two types of divergence and we discuss the transition between these two as a function of parameter choices. The blowup is unavoidable unless the corresponding equations are modified. Our results extend to $3+1$ dimensions.

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J. Eckmann, F. Hassani and H. Zaag
Tue, 3 May 22
51/82

Comments: 19 pages, 5 figures

A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole [CL]

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http://arxiv.org/abs/2204.10620


We develop a Birman-Schwinger principle for the spherically symmetric, asymptotically flat Einstein-Vlasov system. It characterizes stability properties of steady states such as the positive definiteness of an Antonov-type operator or the existence of exponentially growing modes in terms of a one-dimensional variational problem for a Hilbert-Schmidt operator. This requires a refined analysis of the operators arising from linearizing the system, which uses action-angle type variables. For the latter, a single-well structure of the effective potential for the particle flow of the steady state is required. This natural property can be verified for a broad class of singularity-free steady states. As a particular example for the application of our Birman-Schwinger principle we consider steady states where a Schwarzschild black hole is surrounded by a shell of Vlasov matter. We prove the existence of such steady states and derive linear stability if the mass of the Vlasov shell is small compared to the mass of the black hole.

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S. Günther, G. Rein and C. Straub
Mon, 25 Apr 22
25/36

Comments: 58 pages, 1 figure

Accurate Baryon Acoustic Oscillations reconstruction via semi-discrete optimal transport [CEA]

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http://arxiv.org/abs/2110.08868


Optimal transport theory has recently reemerged as a vastly resourceful field of mathematics with elegant applications across physics and computer science. Harnessing methods from geometry processing, we report on the efficient implementation for a specific problem in cosmology — the reconstruction of the linear density field from low redshifts, in particular the recovery of the Baryonic Acoustic Oscillation (BAO) scale. We demonstrate our algorithm’s accuracy by retrieving the BAO scale in noise-less cosmological simulations that are dedicated to cancel cosmic variance; we find uncertainties to be reduced by factor of 4.3 compared with performing no reconstruction, and a factor of 3.1 compared with standard reconstruction.

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S. Hausegger, B. Lévy and R. Mohayaee
Tue, 19 Oct 21
48/98

Comments: Comments welcome! 5 pages excluding references, 2 figures, 1 table

Revisiting the averaged problem in the case of mean-motion resonances of the restricted three-body problem. Global rigorous treatment and application to the co-orbital motion [EPA]

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http://arxiv.org/abs/2106.14810


A classical approach to the restricted three-body problem is to analyze the dynamics of the massless body in the synodic reference frame. A different approach is represented by the perturbative treatment: in particular the averaged problem of a mean-motion resonance allows to investigate the long-term behavior of the solutions through a suitable approximation that focuses on a particular region of the phase space. In this paper, we intend to bridge a gap between the two approaches in the specific case of mean-motion resonant dynamics, establish the limit of validity of the averaged problem, and take advantage of its results in order to compute trajectories in the synodic reference frame. After the description of each approach, we develop a rigorous treatment of the averaging process, estimate the size of the transformation and prove that the averaged problem is a suitable approximation of the restricted three-body problem as long as the solutions are located outside the Hill’s sphere of the secondary. In such a case, a rigorous theorem of stability over finite but large timescales can be proven. We establish that a solution of the averaged problem provides an accurate approximation of the trajectories on the synodic reference frame within a finite time that depend on the minimal distance to the Hill’s sphere of the secondary. The last part of this work is devoted to the co-orbital motion (i.e., the dynamics in 1:1 mean-motion resonance) in the circular-planar case. In this case, an interpretation of the solutions of the averaged problem in the synodic reference frame is detailed and a method that allows to compute co-orbital trajectories is displayed.

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A. Pousse and E. Alessi
Tue, 29 Jun 21
52/101

Comments: 24 pages, 7 figures, 1 table

Blowups and long-time developments of irregularly-shaped Euler-Poisson dominated molecular clouds [CL]

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http://arxiv.org/abs/2102.11550


Motivated by the astrophysical problems of star formations from molecular clouds, we make the first step on the possible behaviors of certain molecular clouds. This article $(1)$ establishes the diffuse boundary problem of Euler-Poisson system for describing the evolution of molecular clouds; $(2)$ proves the local existence, uniqueness and continuation principle of the classical solution to the diffuse boundary problem; $(3)$ proves the classical solution (without any symmetry condition) to the diffuse problem blows up at finite time if there is no the first class of global solution and the data is admissible (large scale, irregularly-shaped, expanding and rotational molecular clouds); $(4)$ proves certain singularities can be removed from the boundary if the data is strongly admissible. This result partially answers Makino’s conjecture $[69]$ on the finite blowup of any tame solution without symmetries and gives the possibilities of star formations, fragmentation and possibilities of formations of shocks and physical vacuum boundary in perfect fluids with Newtonian self-gravity.

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C. Liu
Wed, 24 Feb 21
40/64

Comments: 96 pages

On the existence of linearly oscillating galaxies [CL]

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http://arxiv.org/abs/2102.11672


We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

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M. Hadzic, G. Rein and C. Straub
Wed, 24 Feb 21
42/64

Comments: 104 pages

Linear stability analysis of the homogeneous Couette flow in a 2D isentropic compressible fluid [CL]

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http://arxiv.org/abs/2101.01696


In this paper, we study the linear stability properties of perturbations around the homogeneous Couette flow for a 2D isentropic compressible fluid in the domain $\mathbb{T}\times \mathbb{R}$. In the inviscid case there is a generic Lyapunov type instability for the density and the irrotational component of the velocity field. More precisely, we prove that their $L^2$ norm grows as $t^{1/2}$ and this confirms previous observations in the physics literature. Instead, the solenoidal component of the velocity field experience inviscid damping, meaning that it decays to zero even in the absence of viscosity. For a viscous compressible fluid, we show that the perturbations may have a transient growth of order $\nu^{-1/6}$ (with $\nu^{-1}$ being proportional to the Reynolds number) on a time-scale $\nu^{-1/3}$, after which it decays exponentially fast. This phenomenon is also called enhanced dissipation and our result appears to be the first to detect this mechanism for a compressible fluid, where an exponential decay for the density is not a priori trivial given the absence of dissipation in the continuity equation.

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P. Antonelli, M. Dolce and P. Marcati
Wed, 6 Jan 21
56/82

Comments: 39 pages. A preliminary analysis of the inviscid problem already appeared in our unpublished note arxiv.org/abs/2003.01694

A fast semi-discrete optimal transport algorithm for a unique reconstruction of the early Universe [CEA]

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http://arxiv.org/abs/2012.09074


We leverage powerful mathematical tools stemming from optimal transport theory and transform them into an efficient algorithm to reconstruct the fluctuations of the primordial density field, built on solving the Monge-Amp`ere-Kantorovich equation. Our algorithm computes the optimal transport between an initial uniform continuous density field, partitioned into Laguerre cells, and a final input set of discrete point masses, linking the early to the late Universe. While existing early universe reconstruction algorithms based on fully discrete combinatorial methods are limited to a few hundred thousand points, our algorithm scales up well beyond this limit, since it takes the form of a well-posed smooth convex optimization problem, solved using a Newton method. We run our algorithm on cosmological $N$-body simulations, from the AbacusCosmos suite, and reconstruct the initial positions of $\mathcal{O}(10^7)$ particles within a few hours with an off-the-shelf personal computer. We show that our method allows a unique, fast and precise recovery of subtle features of the initial power spectrum, such as the baryonic acoustic oscillations.

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B. Lévy, R. Mohayaee and S. Hausegger
Thu, 17 Dec 20
62/85

Comments: 22 pages

Structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics [CL]

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http://arxiv.org/abs/1911.06295


We study the structural stability of shock waves and current-vortex sheets in shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time existence and uniqueness of discontinuous solutions satisfying corresponding jump conditions. The equations of SMHD form a symmetric hyperbolic system which is formally analogous to the system of 2D compressible elastodynamics for particular nonphysical deformations. Using this analogy and the recent results in [Morando A., Trakhinin Y., Trebeschi P. Math. Ann. (2019), https://doi.org/10.1007/s00208-019-01920-6] for shock waves in 2D compressible elastodynamics, we prove that shock waves in SMHD are structurally stable if and only if the fluid height increases across the shock front. For current-vortex sheets the fluid height is continuous whereas the tangential components of the velocity and the magnetic field may have a jump. Applying a so-called secondary symmetrization of the symmetric system of SMHD equations, we find a condition sufficient for the structural stability of current-vortex sheets.

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Y. Trakhinin
Fri, 15 Nov 19
37/73

Comments: 18 pages

Axisymmetric solutions in the geomagnetic direction problem [CL]

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http://arxiv.org/abs/1909.11526


The magnetic field outside the earth is in good approximation a harmonic vector field determined by its values at the earth’s surface. The direction problem seeks to determine harmonic vector fields vanishing at infinity and with prescribed direction of the field vector at the surface. In general this type of data does neither guarantee existence nor uniqueness of solutions of the corresponding nonlinear boundary value problem. To determine conditions for existence, to specify the non-uniqueness, and to identify cases of uniqueness is of particular interest when modeling the earth’s (or any other celestial body’s) magnetic field from these data.
Here we consider the case of {\em axisymmetric} harmonic fields $\BB$ outside the sphere $S^2 \subset \real^3$. We introduce a rotation number $\ro \in \ints$ along a meridian of $S^2$ for any axisymmetric H\”older continuous direction field $\DD \neq 0$ on $S^2$ and, moreover, the (exact) decay order $3 \leq \de \in \ints$ of any axisymmetric harmonic field $\BB$ at infinity. Fixing a meridional plane and in this plane $\ro – \de +1 \geq 0$ points $z_n$ (symmetric with respect to the symmetry axis and with $|z_n| > 1$, $n = 1,\ldots,\rho-\de +1$), we prove the existence of an (up to a positive constant factor) unique harmonic field $\BB$ vanishing at $z_n$ and nowhere else, with decay order $\de$ at infinity, and with direction $\DD$ at $S^2$. The proof is based on the global solution of a nonlinear elliptic boundary value problem, which arises from a complex analytic ansatz for the axisymmetric harmonic field in the meridional plane. The coefficients of the elliptic equation are discontinuous and singular at the symmetry axis, which requires solution techniques that are adapted to this special situation.

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R. Kaiser and T. Ramming
Thu, 26 Sep 19
36/61

Comments: N/A

Similarity Inner Solutions for the Pulsar Equation [HEAP]

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http://arxiv.org/abs/1909.08521


Lie symmetries are applied to classify the source of the magnetic field for the Pulsar equation near to the surface of the neutron star. We find that there are six possible different admitted Lie algebras. We apply the corresponding Lie invariants to reduce the Pulsar equation close to the surface to an ordinary differential equation. This equation is solved either with the use of Lie symmetries or the application of the ARS algorithm for singularity analysis to write the analytic solution as a Laurent expansion. These solutions are called inner solutions.

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A. Paliathanasis
Thu, 19 Sep 19
23/71

Comments: 11 pages, 2 figures, accepted for publication by Mathematical Methods in the Applied Science

Numerical treatment of the nonconservative product in a multiscale fluid model for plasmas in thermal nonequilibrium: application to solar physics [CL]

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http://arxiv.org/abs/1806.10436


This contribution deals with the modeling of collisional multicomponent magnetized plasmas in thermal and chemical nonequilibrium aiming at simulating and predicting magnetic reconnections in the chromosphere of the sun. We focus on the numerical simulation of a simplified fluid model in order to properly investigate the influence on shock solutions of a nonconservative product present in the electron energy equation. Then, we derive jump conditions based on travelling wave solutions and propose an original numerical treatment in order to avoid non-physical shocks for the solution, that remains valid in the case of coarse-resolution simulations. A key element for the numerical scheme proposed is the presence of diffusion in the electron variables, consistent with the physically-sound scaling used in the model developed by Graille et al. following a multiscale Chapman-Enskog expansion method [M3AS, 19 (2009) 527–599]. The numerical strategy is eventually assessed in the framework of a solar physics test case. The computational method is able to capture the travelling wave solutions in both the highly- and coarsely-resolved cases.

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Q. Wargnier, S. Faure, B. Graille, et. al.
Thu, 28 Jun 18
49/60

Comments: N/A

The wave equation near flat Friedmann-Lemaître-Robertson-Walker and Kasner Big Bang singularities [CL]

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http://arxiv.org/abs/1805.12558


We consider the wave equation, $\square_g\psi=0$, in fixed flat Friedmann-Lema\^itre-Robertson-Walker and Kasner spacetimes with topology $\mathbb{R}_+\times\mathbb{T}^3$. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface ${t=0}$. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate $L^2$-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the $L^2(\mathbb{T}^3)$ norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

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G. Fournodavlos, A. Franzen and A. Alho
Fri, 1 Jun 18
49/56

Comments: 14 pages, 1 figure

Negative magnetic eddy diffusivity due to oscillatory $α$-effect [CL]

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http://arxiv.org/abs/1711.02390


We study large-scale kinematic dynamo action of steady mirror-antisymmetric flows of incompressible fluid, that involve small spatial scales only, by asymptotic methods of the multiscale stability theory. It turns out that, due to the magnetic $\alpha$-effect in such flows, mean field experiences harmonic oscillations in time on the scale $T_1=\varepsilon t$ without growth or decay. Here $\varepsilon$ is the spatial scale ratio and $t$ is the fast time of the order of the flow turnover time. The interaction of the accompanying fluctuating magnetic field with the flow gives rise to an anisotropic magnetic eddy diffusivity, whose dependence on the direction of the large-scale wave vector generically exhibits a singular behaviour, and thus to negative eddy diffusivity for whichever molecular magnetic diffusivity. Consequently, such flows always act as kinematic dynamos on the time scale $T_2=\varepsilon^2t$. We investigate numerically this dynamo mechanism for two sample flows.

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A. Andrievsky, R. Chertovskih and V. Zheligovsky
Fri, 10 Nov 17
3/55

Comments: N/A

Cosmological Newtonian limits on long time scales [CL]

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http://arxiv.org/abs/1701.03975


We establish the existence of $1$-parameter families of $\epsilon$-dependent solutions to the Einstein-Euler equations with a positive cosmological constant $\Lambda >0$ and a linear equation of state $p=\epsilon^2 K \rho$, $0<K\leq 1/3$, for the parameter values $0<\epsilon < \epsilon_0$. These solutions exist globally to the future, converge as $\epsilon \searrow 0$ to solutions of the cosmological Poison-Euler equations of Newtonian gravity, and are inhomogeneous non-linear perturbations of FLRW fluid solutions.

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C. Liu and T. Oliynyk
Thu, 26 Jan 17
44/68

Comments: 57 pages

Dynamical system modeling fermionic limit [CL]

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http://arxiv.org/abs/1612.05442


The existence of multiple radial solutions to the elliptic equation modeling fermionic cloud of interacting particles is proved for the limiting Planck constant and intermediate values of mass parameters. It is achieved by considering the related nonautonomous dynamical system for which the passage to the limit can be established due to the continuity of the solutions with respect to the parameter going to zero.

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D. Bors and R. Stanczy
Mon, 19 Dec 16
46/54

Comments: 12 pages, 2 figures

Soliton Formation in Neutral Ion Gases: Exact Analysis [CL]

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http://arxiv.org/abs/1209.3077


It is shown here that in neutral ion gases the thermal energy transport can occur in the form of new types of thermal soliton waves. The solitons can form under a vanishing net heating function, and for a quadratic net heating. It is predicted that these solitons play an important role in a diversity of terrestrial and astrophysical phenomena. We claim that the reported soliton waves can be observed under ordinary laboratory conditions.

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B. Mirza
Fri, 29 May 15
34/68

Comments: Figure added, typos corrected

Symmetries of Differential equations and Applications in Relativistic Physics [CL]

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http://arxiv.org/abs/1501.05129


In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new geometric method which relates the point symmetries of the differential equations with the collineations of the underlying manifold where the motion occurs. This geometric method is applied in order the two and three dimensional Newtonian dynamical systems to be classified in relation to the point symmetries; to generalize the Newtonian Kepler-Ermakov system in Riemannian spaces; to study the symmetries between classical and quantum systems and to investigate the geometric origin of the Type II hidden symmetries for the homogeneous heat equation and for the Laplace equation in Riemannian spaces. At last but not least, we apply this geometric approach in order to determine the dark energy models by use the Noether symmetries as a geometric criterion in modified theories of gravity.

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A. Paliathanasis
Thu, 22 Jan 15
2/58

Comments: PhD Thesis; University of Athens (2014); 285 pages N.D.C of Greece this http URL

An Instability of the Standard Model Creates the Anomalous Acceleration Without Dark Energy [CL]

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http://arxiv.org/abs/1412.4001


We introduce a new asymptotic ansatz for spherical perturbations of the Standard Model of Cosmology (SM) which applies during the $p=0$ epoch, and prove that these perturbations trigger instabilities in the SM on the scale of the supernova data. These instabilities create a large, central region of uniform under-density which expands faster than the SM, and this central region of accelerated uniform expansion introduces into the SM {\it precisely} the same range of corrections to redshift vs luminosity as are produced by the cosmological constant in the theory of Dark Energy. A universal behavior is exhibited because all sufficiently small perturbations evolve to a single stable rest point. Moreover, we prove that these perturbations are consistent with, and the instability is triggered by, the one parameter family of self-similar waves which the authors previously proposed as possible time-asymptotic wave patterns for perturbations of the SM at the end of the radiation epoch. Using numerical simulations, we calculate the unique wave in the family that accounts for the same values of the Hubble constant and quadratic correction to redshift vs luminosity as in a universe with seventy percent Dark Energy, $\Omega_{\Lambda}\approx.7$. A numerical simulation of the third order correction associated with that unique wave establishes a testable prediction that distinguishes this theory from the theory of Dark Energy. This explanation for the anomalous acceleration, based on instabilities in the SM together with simple wave perturbations from the radiation epoch that trigger them, provides perhaps the simplest mathematical explanation for the anomalous acceleration of the galaxies that does not invoke Dark Energy.

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J. Smoller, B. Temple and Z. Vogler
Mon, 15 Dec 14
36/53

Comments: N/A

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow [CL]

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http://arxiv.org/abs/1312.6320


It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous), nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. For proving such results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a prize of the French Academy. This formulation leads to simple recurrence relations among the time-Taylor coefficients of the Lagrangian map from initial to current fluid particle positions; the coefficients can then be bounded using elementary methods. We first consider various classes of incompressible fluid flow, governed by the Euler equations, and then turn to a case of compressible flow of cosmological relevance, governed by the Euler-Poisson equations.

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Wed, 25 Dec 13
21/23