A regularizing effect of nonlinear transport equations
Author:
Felix Otto
Journal:
Quart. Appl. Math. 56 (1998), 355-375
MSC:
Primary 35L65; Secondary 35F25, 35Q35, 76S05, 82C70
DOI:
https://doi.org/10.1090/qam/1622511
MathSciNet review:
MR1622511
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Abstract: We consider the semigroup on ${L^1}\left ( {\mathbb {R}^n} \right )$ defined by the nonlinear transport equation for the scalar $s$, \[ {\partial _t}s + div\left ( f\left ( s \right )u \right ) = 0 \qquad in \left ( 0, \infty \right ) \times {\mathbb {R}^n}\] for given velocity field $u$. We show that this nonlinear semigroup is Hölder continuous for $t > 0$ in the uniform operator topology, provided the graph of $f$ has no linear segments. This continuity property—which expresses a regularizing effect of the nonlinearity in the transport equation—is robust with respect to the spatial behaviour of the time-independent velocity field $u$.
P. Bénilan and M. Crandall, Regularizing effects of homogeneous evolution equations, Amer. J. Math., supplement dedicated to P. Hartmann, 1981, pp. 23–30
- Sergio Campanato, Proprietà di inclusione per spazi di Morrey, Ricerche Mat. 12 (1963), 67–86 (Italian). MR 156228
- Kuo Shung Cheng, A regularity theorem for a nonconvex scalar conservation law, J. Differential Equations 61 (1986), no. 1, 79–127. MR 818862, DOI https://doi.org/10.1016/0022-0396%2886%2990126-9
- Joseph G. Conlon, Asymptotic behavior for a hyperbolic conservation law with periodic initial data, Comm. Pure Appl. Math. 32 (1979), no. 1, 99–112. MR 508918, DOI https://doi.org/10.1002/cpa.3160320104
- Michael Crandall and Michel Pierre, Regularizing effects for $u_{t}+A\varphi (u)=0$ in $L^{1}$, J. Functional Analysis 45 (1982), no. 2, 194–212. MR 647071, DOI https://doi.org/10.1016/0022-1236%2882%2990018-0
- Constantine M. Dafermos, Applications of the invariance principle for compact processes. II. Asymptotic behavior of solutions of a hyperbolic conservation law, J. Differential Equations 11 (1972), 416–424. MR 296476, DOI https://doi.org/10.1016/0022-0396%2872%2990055-1
- C. M. Dafermos, Characteristics in hyperbolic conservation laws. A study of the structure and the asymptotic behaviour of solutions, Nonlinear analysis and mechanics: Heriot-Watt Symposium (Edinburgh, 1976), Vol. I, Pitman, London, 1977, pp. 1–58. Res. Notes in Math., No. 17. MR 0481581
- C. M. Dafermos, Regularity and large time behaviour of solutions of a conservation law without convexity, Proc. Roy. Soc. Edinburgh Sect. A 99 (1985), no. 3-4, 201–239. MR 785530, DOI https://doi.org/10.1017/S0308210500014256
- James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
- J. M. Greenberg and Donald D. M. Tong, Decay of periodic solutions of $\partial u/\partial t+\partial f(u)/\partial x=0$, J. Math. Anal. Appl. 43 (1973), 56–71. MR 320488, DOI https://doi.org/10.1016/0022-247X%2873%2990257-6
S. N. Kružkov, First order quasi-linear equations in several independent variables, Math. USSR-Sb. 10, 217–243 (1970)
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI https://doi.org/10.1002/cpa.3160100406
- Pierre-Louis Lions, Benoît Perthame, and Eitan Tadmor, Formulation cinétique des lois de conservation scalaires multidimensionnelles, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 1, 97–102 (French, with English summary). MR 1086510
- Tai-Ping Liu and Michel Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), no. 3, 419–441. MR 735207, DOI https://doi.org/10.1016/0022-0396%2884%2990096-2
O. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. Ser. 2 26, 95–172 (1957)
- Felix Otto, $L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations 131 (1996), no. 1, 20–38. MR 1415045, DOI https://doi.org/10.1006/jdeq.1996.0155
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
- Kevin Zumbrun, Decay rates for nonconvex systems of conservation laws, Comm. Pure Appl. Math. 46 (1993), no. 3, 353–386. MR 1202961, DOI https://doi.org/10.1002/cpa.3160460304
P. Bénilan and M. Crandall, Regularizing effects of homogeneous evolution equations, Amer. J. Math., supplement dedicated to P. Hartmann, 1981, pp. 23–30
S. Campanato, Proprietá di inclusione per spazi di Morrey, Ricerche Mat. 12, 67–86 (1963)
Kuo-Shung Cheng, A regularity theorem for a nonconvex scalar conservation law, J. Differential Equations 61, 79–127 (1986)
J. Conlon, Asymptotic behaviour for a hyperbolic conservation law with periodic initial data, Comm. Pure Appl. Math. 32, 99–112 (1979)
M. Crandall and M. Pierre, Regularizing effects for ${u_t} + A\phi \left ( u \right ) = 0$ in $L^{1}$, J. Funct. Anal. 45, 194–212 (1982)
C. Dafermos, Application of the invariance principle for compact processes II. Asymptotic behaviour of solutions of a hyperbolic conservation law, J. Differential Equations 11, 416–424 (1972)
C. Dafermos, Characteristics in hyperbolic conservation laws, a study of the structure and the asymptotic behaviour of solutions, in Nonlinear Analysis and Mechanics, Vol. 1 (Ed. R. Knops), Pitman, London, 1977
C. Dafermos, Regularity and large time behaviour of solutions of a conservative law without convexity, Proc. Royal Soc. Edin. 99, 201–239 (1985)
J. Glimm and P. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc. 101 (1970)
J. Greenberg and D. Tong, Decay of periodic solutions of $\partial u/\partial t + \partial f\left ( u \right )/\partial x = 0$, J. Math. Anal. Appl. 43, 56–71 (1973)
S. N. Kružkov, First order quasi-linear equations in several independent variables, Math. USSR-Sb. 10, 217–243 (1970)
P. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10, 537–556 (1957)
P.-L. Lions, B. Perthame, and E. Tadmor, Formulation cinétique des lois de conservation scalaires multi-dimensionelles, C. R. Acad. Sci. Paris Sér. I Math. 312, 97–102 (1991)
T. P. Liu and M. Pierre, Source-solutions and asymptotic behaviour in conservation laws, J. Differential Equations 51, 419–441 (1984)
O. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. Ser. 2 26, 95–172 (1957)
F. Otto, $L^{1}$-contraction and uniqueness for quasilinear elliptic-parabolic equations, J. Differential Equations 131, 20–38 (1996) and C. R. Acad. Sci. Paris Sér. I Math. 321, 1005–1010 (1995)
L. Tartar, Compensated compactness and applications to partial differential equations, In Research Notes in Mathematics, vol. 39, Nonlinear Analysis and Mechanics, Heriot-Watt-Sympos., Vol. 4 (R. J. Knops, ed.) Pitman Press, Boston, London, 1975
K. Zumbrun, Decay rates for nonconvex systems of conservation laws, Comm. Pure Appl. Math. 46, 353–386 (1993)
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© Copyright 1998
American Mathematical Society
