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Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws

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Transport Equations and Multi-D Hyperbolic Conservation Laws

Part of the book series: Lecture Notes of the Unione Matematica Italiana ((UMILN,volume 5))

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References

  1. L. Ambrosio, C. De Lellis, and J. Maly. On the chain rule for the divergence of BV like vector fields: Applications, partial results, open problems. In Perspectives in Nonlinear Partial Differential Equations: in honor of Haim Brezis. Birkhäuser, 2006.

    Google Scholar 

  2. L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000.

    MATH  Google Scholar 

  3. M. Bézard. Régularité L p précisée des moyennes dans les équations de transport. Bull. Soc. Math. France, 122(1):29–76, 1994.

    MATH  MathSciNet  Google Scholar 

  4. F. Bouchut. Hypoelliptic regularity in kinetic equations. J. Math. Pures Appl. (9), 81(11):1135–1159, 2002.

    MATH  MathSciNet  Google Scholar 

  5. F. Bouchut and F. James. Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness. Comm. Partial Differential Equations, 24(11–12):2173–2189, 1999.

    MATH  MathSciNet  Google Scholar 

  6. Y. Brenier. Averaged multivalued solutions for scalar conservation laws. SIAM J. Numer. Anal., 21(6):1013–1037, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  7. G.-Q. Chen and H. Frid. Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal., 147(2):89–118, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  8. G.-Q. Chen and M. Rascle. Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Ration. Mech. Anal., 153(3):205–220, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  9. K. S. Cheng. A regularity theorem for a nonconvex scalar conservation law. J. Differential Equations, 61(1):79–127, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  10. C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000.

    MATH  Google Scholar 

  11. C. De Lellis, F. Otto, and M. Westdickenberg. Structure of entropy solutions for multi-dimensional scalar conservation laws. Arch. Ration. Mech. Anal., 170(2):137–184, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. De Lellis and T. Rivière. The rectifiability of entropy measures in one space dimension. J. Math. Pures Appl. (9), 82(10):1343–1367, 2003.

    MATH  MathSciNet  Google Scholar 

  13. R. J. DiPerna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. of Math. (2), 130(2):321–366, 1989.

    Article  MathSciNet  Google Scholar 

  14. R. J. DiPerna, P.-L. Lions, and Y. Meyer. L p regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(3–4):271–287, 1991.

    MATH  MathSciNet  Google Scholar 

  15. W. Gautschi. On inverses of Vandermonde and confluent Vandermonde matrices. Numer. Math., 4:117–123, 1962.

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 76(1):110–125, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Hoff. The sharp form of Oleĭnik’s entropy condition in several space variables. Trans. Amer. Math. Soc., 276(2):707–714, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  18. P.-E. Jabin and B. Perthame. Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim. Calc. Var., 8:761–774 (electronic), 2002.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. N. Kružkov. First order quasilinear equations with several independent variables. Mat. Sb. (N.S.), 81 (123):228–255, 1970.

    MathSciNet  Google Scholar 

  20. P.-L. Lions, B. Perthame, and E. Tadmor. A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc., 7(1):169–191, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  21. P. Mattila. Geometry of sets and measures in Euclidean spaces, volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.

    MATH  Google Scholar 

  22. O. A. Oleĭnik. Discontinuous solutions of non-linear differential equations. Uspehi Mat. Nauk (N.S.), 12(3(75)):3–73, 1957.

    Google Scholar 

  23. E. Yu. Panov. Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ., 2(4):885–908, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  24. E. Tadmor and T. Tao. Velocity averaging, kinetic formulations, and regularizing effects in quasilinear PDEs. Comm. Pure Appl. Math., 2006.

    Google Scholar 

  25. A. Vasseur. Strong traces for solutions of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal., 160(3):181–193, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Westdickenberg. Some new velocity averaging results. SIAM J. Math. Anal., 33(5):1007–1032 (electronic), 2002.

    Article  MATH  MathSciNet  Google Scholar 

  27. K. Zumbrun. Decay rates for nonconvex systems of conservation laws. Comm. Pure Appl. Math., 46(3):353–386, 1993.

    Article  MATH  MathSciNet  Google Scholar 

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Authors
  1. Gianluca Crippa
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  2. Felix Otto
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  3. Michael Westdickenberg
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Crippa, G., Otto, F., Westdickenberg, M. (2008). Regularizing Effect of Nonlinearity in Multidimensional Scalar Conservation Laws. In: Transport Equations and Multi-D Hyperbolic Conservation Laws. Lecture Notes of the Unione Matematica Italiana, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76781-7_3

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