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Continuity of the flow map for symmetric hyperbolic systems and its application to the Euler-Poisson system


We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right-hand sides in different functional setting, including weighted Sobolev spaces Hs δ. An essential tool to achieve the continuity of the flow map is a new type of energy estimate, which we call a low regularity energy estimate. We then apply these results to the Euler-Poisson system which describes various systems of physical interest.

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  1. [BadV92]

    H. Beirão da Veiga, Data dependence in the mathematical theory of compressible inviscid fluids, Arch. Rational Mech. Anal. 119 (1992), 109–127.

    MathSciNet  Article  Google Scholar 

  2. [BCD11]

    H. Bahouri, J. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Springer, Heidelberg-London-New York, 2011.

    Book  Google Scholar 

  3. [BK14]

    U. Brauer and L. Karp, Local existence of solutions of self gravitating relativistic perfect fluids, Comm. Math. Phys. 325 (2014), 105–141.

    MathSciNet  Article  Google Scholar 

  4. [BK15]

    U. Brauer and L. Karp, Elliptic equations in weighted Besov spaces on asymptotically flat Riemannian manifolds, Manuscripta Math. 148 (2015), 59–97.

    MathSciNet  Article  Google Scholar 

  5. [BK18]

    U. Brauer and L. Karp, Local existence of solutions to the Euler-Poisson system, including densities without compact support, J. Differential Equations 264 (2018), 755–785.

    MathSciNet  Article  Google Scholar 

  6. [Bra92]

    U. Brauer, An existence theorem for perturbed newtonian cosmological models, J. Math. Phys. 33 (1992), 1224–1233.

    MathSciNet  Article  Google Scholar 

  7. [Can75]

    M. Cantor, Spaces of functions with asymptotic conditions onn, Indiana Univ. Math. J. 24 (1975), 897–902.

    MathSciNet  Article  Google Scholar 

  8. [CBC81]

    Y. Choquet-Bruhat and D. Christodoulou, Elliptic systems in Hs,δ spaces on manifolds which are euclidian at infinity, Acta Math. 146 (1981), 129–150.

    MathSciNet  Article  Google Scholar 

  9. [Chr00]

    D. Christodoulou, The Action Principle and Partial Differential Equations, Princeton University Press, Princeton, NJ, 2000.

    Book  Google Scholar 

  10. [FM72]

    A. E. Fischer and J. E. Marsden, The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system, Comm. Math. Phys. 28 (1972), 1–38.

    MathSciNet  Article  Google Scholar 

  11. [Kat75a]

    T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal. 58 (1975), 181–205.

    MathSciNet  Article  Google Scholar 

  12. [Kat75b]

    T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Spectral Theory and Differential Equations, Springer, Berlin, 1975, pp. 25–70.

    Chapter  Google Scholar 

  13. [Kat03]

    D. Kateb, On the boundedness of the mapping f → |f|µ, µ 1 on Besov spaces, Math. Nachr. 248/249 (2003), 110–128.

    MathSciNet  Article  Google Scholar 

  14. [KL84]

    T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal. 56 (1984), 15–28.

    MathSciNet  Article  Google Scholar 

  15. [KP88]

    T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907.

    MathSciNet  Article  Google Scholar 

  16. [LP87]

    J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), 37–91.

    MathSciNet  Article  Google Scholar 

  17. [Maj84]

    A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer, New York, 1984.

    Book  Google Scholar 

  18. [Mak86]

    T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in Patterns and Waves, North Holland, Amsterdam, 1986, pp. 459–479.

    Google Scholar 

  19. [Max06]

    D. Maxwell, Rough solutions of the Einstein constraint equations, J. Reine Angew. Math. 590 (2006), 1–29.

    MathSciNet  Article  Google Scholar 

  20. [NW73]

    L. Nirenberg and H. Walker, The null spaces of elliptic differential operators inn, J. Math. Anal. Appl. 42 (1973), 271–301.

    MathSciNet  Article  Google Scholar 

  21. [Rau12]

    J. Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, American Mathematical Society, Providence, RI, 2012.

    Book  Google Scholar 

  22. [Rin78]

    W. Rindler, Essential Relativity, Springer, Berlin-Heidelberg, 1978.

    MATH  Google Scholar 

  23. [RR94]

    G. Rein and A. D. Rendall, Global existence ofclassical solutions to the Vlasov-Poisson system in a three-dimensional, cosmological setting, Arch. Rational Mech. Anal. 126 (1994), 183–201.

    MathSciNet  Article  Google Scholar 

  24. [RS96]

    T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin, 1996.

    Book  Google Scholar 

  25. [Spe09]

    J. Speck, Well-posedness for the Euler-Nordstrom system with cosmological constant, J. Hyperbolic Differ. Equ. 6 (2009), 313–358.

    MathSciNet  Article  Google Scholar 

  26. [Ste70]

    E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

  27. [Tao06]

    T. Tao, Nonlinear Dispersive Equations, American Mathematical Society, Providence, RI, 2006.

    Book  Google Scholar 

  28. [Tay91]

    M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhauser, Boston, MA, 1991.

    Book  Google Scholar 

  29. [Tay97]

    M. Taylor, Partial Differential Equations. vol III. Nonlinear Equations, Springer, New York, 1997.

    Google Scholar 

  30. [Tem79]

    R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  31. [Tri76]

    H. Triebel, Spaces of Kudrjavcev type I. Interpolation, embedding, and structure, J. Math. Anal. Appl. 56 (1976), 253–277.

    MathSciNet  Article  Google Scholar 

  32. [Tzv06]

    N. Tzvetkov, Ill-posedness issues for nonlinear dispersive equations, in Lectures on Nonlinear Dispersive Equations, Gakkbōtosho, Tokyo, 2006, pp. 63–103.

    MATH  Google Scholar 

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Corresponding author

Correspondence to Lavi Karp.

Additional information

Dedicated to Lawrence Zalcman

U. B. gratefully acknowledges support from Grant MTM2016-75465 and Grant PID2019-103860GB-I00 by MINECO, Spain and UCM-GR17-920894.

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Brauer, U., Karp, L. Continuity of the flow map for symmetric hyperbolic systems and its application to the Euler-Poisson system. JAMA 141, 113–163 (2020).

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