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Archive for Rational Mechanics and Analysis

, Volume 226, Issue 2, pp 809–849 | Cite as

Statistical Solutions of Hyperbolic Conservation Laws: Foundations

  • U. S. Fjordholm
  • S. Lanthaler
  • S. Mishra
Article

Abstract

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on p-integrable functions. To do so, we prove the equivalence between probability measures on L p spaces and infinite families of correlation measures. Each member of this family, termed a correlation marginal, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a L p space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions, even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting entropy statistical solutions exist, are unique and are stable with respect to the 1-Wasserstein metric on probability measures on L 1.

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References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Basel (2005)Google Scholar
  2. 2.
    Ball, J.: A version of the fundamental theorem for Young measures. In: Rascle, M., Serre, D., Slemrod, M. (eds.) PDEs and Continuum Models of Phase Transitions. Lecture Notes in Physics, vol. 344, pp. 207–215. Springer, Berlin (1989)Google Scholar
  3. 3.
    Bardos C., Titi E., Wiedemann E.: The vanishing viscosity as a selection principle for the Euler equations: the case of 3D shear flow. C.R. Math. Acad. Sci. Paris 350(15–16), 757–760 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benzoni-Gavage, S., Serre, D.: Multidimensional Hyperbolic Partial Differential Equations. First Order Systems and Applications. Oxford University Press, Oxford (2007)Google Scholar
  5. 5.
    Bertoin J.: The inviscid Burgers equation with Brownian initial velocity. Commun. Math. Phys. 193(2), 397–406 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bianchini S., Bressan A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161(1), 223–342 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bijl, H., Lucor, D., Mishra, S., Schwab, Ch. (eds.): Uncertainty quantification in computational fluid dynamics. Lecture Notes in Computational Science and Engineering, vol. 92, Springer, Berlin (2014)Google Scholar
  8. 8.
    Bressan, A.: Hyperbolic Systems of Conservation Laws: The One Dimensional Cauchy Problem. Oxford University Press, Oxford (2000)Google Scholar
  9. 9.
    Carraro L., Duchon J.: Intrinsic statistical solutions of the Burgers equation and Levy processes. C. R. Math. Acad. Sci. Paris 319(8), 855–858 (1994)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Carraro L., Duchon J.: Burgers equation with initial conditions with homogeneous and independent increments. Ann. Inst. H. Poincaré Anal. Non Lineare, 15(4), 431–458 (1998)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Chae D.: The vanishing viscosity limit of statistical solutions of the Navier–Stokes equations. I. 2-D periodic case. J. Math. Anal. Appl., 155(2), 437–459 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chae D.: The vanishing viscosity limit of statistical solutions of the Navier–Stokes equations. II. The general case. J. Math. Anal. Appl., 155(2), 460–484 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Comon, Pierre, Golub, Gene, Lim, Lek-Heng, Mourrain, Bernard: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl, 30(3): 1254–1279 (2008)Google Scholar
  14. 14.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Oxford (1992)Google Scholar
  15. 15.
    Dafermos, C.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2000)Google Scholar
  16. 16.
    De Lellis, C., Székelyhidi Jr. L.: The Euler equations as a differential inclusion. Ann. Math. 170(3), 1417–1436 (2009)Google Scholar
  17. 17.
    Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Math. 68(7), 1157–1190 (2015)Google Scholar
  18. 18.
    Demoulini S., Stuart D. M. A., Tzavaras A. E.: Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. Arch. Ration. Mech. Anal. 205(3), 927–961 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Diestel, J., Uhl, J. J.: Vector Measures. American Mathematical Society, Providence (1977)Google Scholar
  20. 20.
    DiPerna R. J.: Measure-valued solutions to conservation laws. Arch. Ration. Mech. Anal. 88, 223–270 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    DiPerna R. J., Majda A.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Edwards, R. E.: Functional Analysis. Theory and Applications. Holt, Rinehart and Winston, Inc. (1965)Google Scholar
  23. 23.
    Fjordholm U. S., Käppeli R., Mishra S., Tadmor E.: Construction of approximate entropy measure-valued solutions for hyperbolic systems of conservation laws. Found. Comput. Math. 17(3), 763–827 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Fjordholm U. S., Mishra S., Tadmor E.: On the computation of measure-valued solutions. Acta Numer. 25, 567–679 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Fjordholm, U. S., Lye, K. O., Mishra, S.: Statistical solutions of hyperbolic conservation laws II: numerical approximation in the scalar case. In preparation (2017)Google Scholar
  26. 26.
    Fjordholm, U. S., Lye, K. O., Mishra, S., Weber, F. R.: Statistical solutions of hyperbolic conservation laws III: numerical approximation for multi-dimensional systems. In preparation (2017).Google Scholar
  27. 27.
    Foiaş C.: Statistical study of Navier–Stokes equations I. Rend. Sem. Mat. Univ. Padova 48, 219–348 (1972)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Foiaş C.: Statistical study of Navier–Stokes equations II. Rend. Sem. Mat. Univ. Padova 49, 9–123 (1973)zbMATHGoogle Scholar
  29. 29.
    Foiaş, C., Manley, O., Rosa, R., Temam, R.: Navier–Stokes Equations and Turbulence. Cambridge University Press, Cambridge (2001)Google Scholar
  30. 30.
    Frisch, U.: Turbulence. Cambridge University Press, Cambridge (1995)Google Scholar
  31. 31.
    Ghanem, R., Higdon, D., Owhadi, H. (eds.): Handbook of Uncertainty Quantification, Springer, Berlin (2016)Google Scholar
  32. 32.
    Glimm J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18(4), 697–715 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Grafakos, L.: Classical Fourier Analysis. Springer, New York (2008)Google Scholar
  34. 34.
    Gwiazda P., Swierczewska-Gwiazda A., Wiedemann E.: Weak-Strong uniqueness for measure-valued solutions of some compressible fluid models. Nonlinearity 28, 3873–3890 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Holden, H., Risebro, N. H.: Front Tracking for Hyperbolic Conservation Laws. Springer, Berlin (2011)Google Scholar
  36. 36.
    Illner R., Wick J.: On statistical and measure-valued solutions of differential equations. J. Math. Anal. Appl., 157(2), 351–365 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Isserlis L.: On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12, 134–139 (1918)CrossRefGoogle Scholar
  38. 38.
    Klenke, A.: Probability Theory. A Comprehensive Course. 2nd edn., Springer, London (2014)Google Scholar
  39. 39.
    Kruzkov S. N.: First order quasilinear equations in several independent variables. Math USSR SB, 10(2), 217–243 (1970)CrossRefGoogle Scholar
  40. 40.
    Lim H., Yu Y., Glimm J., Li X. L., Sharp D. H.: Chaos, transport and mesh convergence for fluid mixing. Acta Math. Appl. Sin., 24(3), 355–368 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Panov, E. Yu.: On the statistical solutions of the Cauchy problem for a first-order quasilinear equation (Russian). Mat. Model., 14(3), 17–26 (2002)Google Scholar
  42. 42.
    Rasmussen, C. E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2006). http://www.gaussianprocess.org/gpml/chapters/
  43. 43.
    Schochet S.: Examples of measure-valued solutions. Commun. Partial Differ. Equ. 14(5), 545–575 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Stuart A. M.: Inverse problems: a Bayesian perspective. Acta Num. 19, 451–559 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. vol. 58. American Mathematical Society, Providence (2003)Google Scholar
  46. 46.
    Øksendal, B.: Stochastic Differential Equations. An Introduction with Applications. 6th edn., Springer, Berlin (2003)Google Scholar

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  2. 2.Swiss Plasma Center, SB SPC-THLausanneSwitzerland
  3. 3.Seminar for Applied MathematicsZürichSwitzerland

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