Duke Mathematical Journal

On the Lagrangian structure of transport equations: The Vlasov–Poisson system

Luigi Ambrosio, Maria Colombo, and Alessio Figalli

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

The Vlasov–Poisson system is an important nonlinear transport equation, used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions d3 under strong assumptions on the initial data, whereas weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles. In this article we develop several general tools concerning the Lagrangian structure of transport equations with nonsmooth vector fields, and we apply these results to show that weak/renormalized solutions of Vlasov–Poisson are Lagrangian and actually that the concepts of renormalized and Lagrangian solutions are equivalent. As a corollary, we prove that finite-energy solutions in dimension d4 are transported by a global flow (in particular, they preserve all the natural Casimir invariants), and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data.

Article information

Source
Duke Math. J., Volume 166, Number 18 (2017), 3505-3568.

Dates
Received: 29 April 2016
Revised: 7 May 2017
First available in Project Euclid: 8 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1504836225

Digital Object Identifier
doi:10.1215/00127094-2017-0032

Mathematical Reviews number (MathSciNet)
MR3732882

Zentralblatt MATH identifier
06837466

Subjects
Primary: 35F25: Initial value problems for nonlinear first-order equations
Secondary: 35Q83: Vlasov-like equations 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 37C10: Vector fields, flows, ordinary differential equations

Keywords
Vlasov–Poisson transport equations Lagrangian flows renormalized solutions

Citation

Ambrosio, Luigi; Colombo, Maria; Figalli, Alessio. On the Lagrangian structure of transport equations: The Vlasov–Poisson system. Duke Math. J. 166 (2017), no. 18, 3505--3568. doi:10.1215/00127094-2017-0032. https://projecteuclid.org/euclid.dmj/1504836225


Export citation

References

  • [1] L. Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math. 158 (2004), 227–260.
  • [2] L. Ambrosio, M. Colombo, and A. Figalli, Existence and uniqueness of maximal regular flows for non-smooth vector fields. Arch. Ration. Mech. Anal. 218 (2015), 1043–1081.
  • [3] L. Ambrosio and G. Crippa, “Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields” in Transport Equations and Multi-$D$ Hyperbolic Conservation Laws, Lect. Notes Unione Mat. Ital. 5, Springer, Berlin, 2008, 3–57.
  • [4] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Wasserstein Space of Probability Measures, 2nd ed., Lect. Math. ETH Zürich, Birkhäuser, Basel, 2008.
  • [5] A. A. Arsen’ev, Existence in the large of a weak solution of Vlasov’s system of equations (in Russian), Ž. Vyčisl. Mat. i Mat. Fiz. 15 (1975), 136–147, 276.
  • [6] C. Bardos and P. Degond, Global existence for the Vlasov–Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), 101–118.
  • [7] J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations 25 (1977), 342–364.
  • [8] A. Bohun, F. Bouchut, and G. Crippa, Lagrangian flows for vector fields with anisotropic regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), 1409–1429.
  • [9] A. Bohun, F. Bouchut, and G. Crippa, Lagrangian solutions to the Vlasov–Poisson equation with $L^{1}$ density, J. Differential Equations 260 (2016), 3576–3597.
  • [10] F. Bouchut and G. Crippa, “Équations de transport à coefficient dont le gradient est donné par une intégrale singulière” in Séminaire: Équations aux Dérivées Partielles, 2007–2008, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2009, no. I.
  • [11] F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperbolic Differ. Equ. 10 (2013), 235–282.
  • [12] M. Colombo, Flows of non-smooth vector fields and degenerate elliptic equations, Ph.D. dissertation, Scuola Normale Superiore di Pisa, Pisa, 2015.
  • [13] G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna–Lions flow, J. Reine Angew. Math. 616 (2008), 15–46.
  • [14] R. J. DiPerna and P.-L. Lions, Solutions globales d’équations du type Vlasov–Poisson, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 655–658.
  • [15] R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov–Maxwell systems, Comm. Pure Appl. Math. 42 (1989), 729–757.
  • [16] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), 511–547.
  • [17] R. L. Dobrushin, Vlasov equations (in Russian), Funktsional. Anal. i Prilozhen. 13 (1979), 48–58, 96.
  • [18] I. Gasser, P.-E. Jabin, and B. Perthame, Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 1259–1273.
  • [19] E. Horst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation, I: General theory, Math. Methods Appl. Sci. 3 (1981), 229–248; II: Special cases, 4 (1982), 19–32.
  • [20] E. Horst and R. Hunze, Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Methods Appl. Sci. 6 (1984), 262–279.
  • [21] R. Illner and H. Neunzert, An existence theorem for the unmodified Vlasov equation, Math. Methods Appl. Sci. 1 (1979), 530–544.
  • [22] S. V. Iordanskiĭ, The Cauchy problem for the kinetic equation of plasma (in Russian), Trudy Mat. Inst. Steklov. 60 (1961), 181–194.
  • [23] J. Kristensen and G. Mingione, The singular set of minima of integral functionals, Arch. Ration. Mech. Anal. 180 (2006), 331–398.
  • [24] C. Le Bris and P.-L. Lions, Renormalized solutions of some transport equations with partially $W^{1,1}$ velocities and applications, Ann. Mat. Pura Appl. (4) 183 (2003), 97–130.
  • [25] P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov–Poisson system, Invent. Math. 105 (1991), 415–430.
  • [26] G. Loeper, Uniqueness of the solution to the Vlasov–Poisson system with bounded density, J. Math. Pures Appl. (9) 86 (2006), 68–79.
  • [27] C. Pallard, Space moments of the Vlasov–Poisson system: Propagation and regularity, SIAM J. Math. Anal. 46 (2014), 1754–1770.
  • [28] B. Perthame, Time decay, propagation of low moments and dispersive effects for kinetic equations, Comm. Partial Differential Equations 21 (1996), 659–686.
  • [29] K. Pfaffelmoser, Global classical solutions of the Vlasov–Poisson system in three dimensions for general initial data, J. Differential Equations 95 (1992), 281–303.
  • [30] G. Rein, “Collisionless kinetic equations from astrophysics—The Vlasov–Poisson system” in Handbook of Differential Equations: Evolutionary Equations, Vol. III, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, 383–476.
  • [31] J. Schaeffer, Global existence for the Poisson–Vlasov system with nearly symmetric data, J. Differential Equations 69 (1987), 111–148.
  • [32] S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov’s equation, Osaka J. Math. 15 (1978), 245–261.
  • [33] S. Wollman, Global-in-time solutions of the two-dimensional Vlasov–Poisson systems, Comm. Pure Appl. Math. 33 (1980), 173–197.
  • [34] X. Zhang and J. Wei, The Vlasov–Poisson system with infinite kinetic energy and initial data in $L^{p}(\mathbb{R}^{6})$, J. Math. Anal. Appl. 341 (2008), 548–558.