메뉴 건너뛰기




Volumn 41, Issue 4, 2004, Pages 409-438

A survey of entropy methods for partial differential equations

Author keywords

[No Author keywords available]

Indexed keywords


EID: 13844271552     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-04-01032-8     Document Type: Article
Times cited : (61)

References (56)
  • 2
    • 84860086328 scopus 로고    scopus 로고
    • Undercompressive waves in driven thin film flow: Theory, computation, and experiment
    • [B-M-S], AMS/IP Stud. Adv. Math, AMS. MR 1708749
    • [B-M-S] A. Bertozzi, A. Münch and M. Shearer, Undercompressive waves in driven thin film flow: theory, computation, and experiment, in Trends in Mathematical Physics AMS/IP Stud. Adv. Math, AMS, 1999. MR 1708749
    • (1999) Trends in Mathematical Physics
    • Bertozzi, A.1    Münch, A.2    Shearer, M.3
  • 4
    • 0034339675 scopus 로고    scopus 로고
    • Derivation of maximum entropy principles in two-dimensional turbulence via large deviations
    • [B-E-T]. MR 1751700 (2001a:76087)
    • [B-E-T] C. Boucher, R. Ellis and B. Turkington, Derivation of maximum entropy principles in two-dimensional turbulence via large deviations, J. Stat. Physics 98 (2000), 1235-1278. MR 1751700 (2001a:76087)
    • (2000) J. Stat. Physics , vol.98 , pp. 1235-1278
    • Boucher, C.1    Ellis, R.2    Turkington, B.3
  • 5
    • 0037506301 scopus 로고
    • The motion of a surface by its mean curvature
    • [B], Princeton University Press. MR 0485012 (82c:49035)
    • [B] K. Brakke, The Motion of a Surface by its Mean Curvature, Mathematical Notes 20, Princeton University Press, 1978. MR 0485012 (82c:49035)
    • (1978) Mathematical Notes , vol.20
    • Brakke, K.1
  • 6
    • 84990576577 scopus 로고
    • Partial regularity of suitable weak solutions of the Navier-Stokes equations
    • [C-K-N]. MR 0673830 (84m:35097)
    • [C-K-N] L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771-831. MR 0673830 (84m:35097)
    • (1982) Comm. Pure Appl. Math. , vol.35 , pp. 771-831
    • Caffarelli, L.1    Kohn, R.2    Nirenberg, L.3
  • 8
    • 18044401055 scopus 로고    scopus 로고
    • Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities
    • [C-J-M]. MR 1853037 (2002j:35188)
    • [C-J-M] J. Carrillo, A. Jungel, P. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001), 1-82. MR 1853037 (2002j:35188)
    • (2001) Monatsh. Math. , vol.133 , pp. 1-82
    • Carrillo, J.1    Jungel, A.2    Markowich, P.3    Toscani, G.4    Unterreiter, A.5
  • 9
    • 34250963757 scopus 로고
    • The thermodynamics of elastic materials with heat conduction and viscosity
    • [C-N]. MR 0153153 (27:3122)
    • [C-N] B. Coleman and W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Rat. Mech. Analysis 13 (1963), 167-178. MR 0153153 (27:3122)
    • (1963) Arch. Rat. Mech. Analysis , vol.13 , pp. 167-178
    • Coleman, B.1    Noll, W.2
  • 10
    • 0019664885 scopus 로고
    • The second law of thermodynamics for systems with approximate cycles
    • [C-O-S]. MR 0633211 (83e:80002)
    • [C-O-S] B. Coleman, D. Owen and J. Serrin, The second law of thermodynamics for systems with approximate cycles, Arch. Rat. Mech. Analysis 77 (1981), 103-142. MR 0633211 (83e:80002)
    • (1981) Arch. Rat. Mech. Analysis , vol.77 , pp. 103-142
    • Coleman, B.1    Owen, D.2    Serrin, J.3
  • 11
    • 84968507121 scopus 로고
    • Some properties of viscosity solutions of Hamilton-Jacobi equations
    • [C-E-L]. MR 0732102 (86a:35031)
    • [C-E-L] M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. AMS 282 (1984), 487-502. MR 0732102 (86a:35031)
    • (1984) Trans. AMS , vol.282 , pp. 487-502
    • Crandall, M.G.1    Evans, L.C.2    Lions, P.L.3
  • 12
    • 84967758647 scopus 로고
    • Viscosity solutions of Hamilton-Jacobi equations
    • [C-L]. MR 0690039 (85g:35029)
    • [C-L] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. AMS 277 (1983), 1-42. MR 0690039 (85g:35029)
    • (1983) Trans. AMS , vol.277 , pp. 1-42
    • Crandall, M.G.1    Lions, P.L.2
  • 13
    • 13844253011 scopus 로고
    • Entropy and partial differential equations
    • [D], Longman. MR 1413296 (97h:35179)
    • [D] W. Day, Entropy and Partial Differential Equations, Pitman Research Notes in Mathematics Series 295, Longman, 1993. MR 1413296 (97h:35179)
    • (1993) Pitman Research Notes in Mathematics Series , vol.295
    • Day, W.1
  • 14
    • 13844265424 scopus 로고
    • Efficiency and existence of entropy in classical thermodynamics
    • [D-S]. MR 0449196 (56:7501)
    • [D-S] W. Day and M. Šilhavý, Efficiency and existence of entropy in classical thermodynamics, Arch. Rat. Mech. Analysis 66 (1977), 73-81. MR 0449196 (56:7501)
    • (1977) Arch. Rat. Mech. Analysis , vol.66 , pp. 73-81
    • Day, W.1    Šilhavý, M.2
  • 16
    • 0035940508 scopus 로고    scopus 로고
    • Statistical equilibrium predictions of jets and spots on Jupiter
    • [D-H-M-T]. MR 1858819 (2002h:85004)
    • [D-H-M-T] M. DiBattista, K. Haven, A. Majda and B. Turkington, Statistical equilibrium predictions of jets and spots on Jupiter, Proc. Nat. Acad. Sciences 96 (2001), 12346-12350. MR 1858819 (2002h:85004)
    • (2001) Proc. Nat. Acad. Sciences , vol.96 , pp. 12346-12350
    • Dibattista, M.1    Haven, K.2    Majda, A.3    Turkington, B.4
  • 22
    • 84971110219 scopus 로고
    • The perturbed test function method for viscosity solutions of nonlinear PDE
    • [E2]. MR 1007533 (91c:35017)
    • [E2] L. C. Evans, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Royal Soc. Edinburgh 111 (1989), 359-375. MR 1007533 (91c:35017)
    • (1989) Proc. Royal Soc. Edinburgh , vol.111 , pp. 359-375
    • Evans, L.C.1
  • 24
    • 0013123466 scopus 로고
    • Thermodynamics based on the Hahn-Banach theorem: The Clausius inequality
    • [F-L]. MR 0690294 (84f:80001)
    • [F-L] M. Feinberg and R. Lavine, Thermodynamics based on the Hahn-Banach Theorem: the Clausius inequality, Arch. Rat. Mech. Analysis 82 (1983), 203-293. MR 0690294 (84f:80001)
    • (1983) Arch. Rat. Mech. Analysis , vol.82 , pp. 203-293
    • Feinberg, M.1    Lavine, R.2
  • 25
    • 0043046940 scopus 로고    scopus 로고
    • Probabilistic approach to the small viscosity asymptotics for Navier-Stokes equations
    • [F]. MR 1603550 (99g:76033)
    • [F] M. I. Freidlin, Probabilistic approach to the small viscosity asymptotics for Navier-Stokes equations, Nonlinear Analysis 30 (1997), 4069-4076. MR 1603550 (99g:76033)
    • (1997) Nonlinear Analysis , vol.30 , pp. 4069-4076
    • Freidlin, M.I.1
  • 28
    • 0002506597 scopus 로고
    • An axiomatic foundation for continuum thermodynamics
    • [G-W]. MR 0214335 (35:5186)
    • [G-W] M. Gurtin and W. Williams, An axiomatic foundation for continuum thermodynamics, Arch. Rat. Mech. Analysis 26 (1967), 83-117. MR 0214335 (35:5186)
    • (1967) Arch. Rat. Mech. Analysis , vol.26 , pp. 83-117
    • Gurtin, M.1    Williams, W.2
  • 29
    • 0002950929 scopus 로고
    • Shock waves and entropy
    • [Lx1], Academic Press. MR 0393870 (52:14677)
    • [Lx1] P. Lax, Shock waves and entropy, in Contributions to Nonlinear Functional Analysis, Academic Press (1971), 603-634. MR 0393870 (52:14677)
    • (1971) Contributions to Nonlinear Functional Analysis , pp. 603-634
    • Lax, P.1
  • 30
    • 0039815296 scopus 로고
    • Shock waves, increase of entropy and loss of information
    • [Lx2], ed. by S. S. Chern, MSRI Publications, Springer. MR 0765233 (86b:35130)
    • [Lx2] P. Lax, Shock waves, increase of entropy and loss of information, in Seminar on Nonlinear PDE, ed. by S. S. Chern, MSRI Publications, Springer (1984), 129-171. MR 0765233 (86b:35130)
    • (1984) Seminar on Nonlinear PDE , pp. 129-171
    • Lax, P.1
  • 31
    • 0007133108 scopus 로고    scopus 로고
    • A guide to entropy and the second law of thermodynamics
    • [L-Y]. MR 1616141 (99c:80002)
    • [L-Y] E. Leib and J. Yngvason, A guide to entropy and the second law of thermodynamics, Notices AMS 45 (1998), 571-581. MR 1616141 (99c:80002)
    • (1998) Notices AMS , vol.45 , pp. 571-581
    • Leib, E.1    Yngvason, J.2
  • 32
    • 0000324115 scopus 로고
    • On the parabolic kernel of the Schrödinger operator
    • [L-Ya]. MR 0834612 (87f:58156)
    • [L-Ya] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math. 156 (1986), 153-201. MR 0834612 (87f:58156)
    • (1986) Acta. Math. , vol.156 , pp. 153-201
    • Li, P.1    Yau, S.T.2
  • 33
    • 0032344639 scopus 로고    scopus 로고
    • A new proof of the Caffarelli-Kohn-Nirenberg theorem
    • [Li]. MR 1488514 (98k:35151)
    • [Li] F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg Theorem, Comm. Pure and Applied Math. 51 (1998), 241-257. MR 1488514 (98k:35151)
    • (1998) Comm. Pure and Applied Math. , vol.51 , pp. 241-257
    • Lin, F.1
  • 35
    • 0030539239 scopus 로고    scopus 로고
    • Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrandian coordinates
    • [L-P-S]. MR 1383202 (97e:35107)
    • [L-P-S] P. L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrandian coordinates, Comm. Pure Appl. Math. 49 (1996), 599-638. MR 1383202 (97e:35107)
    • (1996) Comm. Pure Appl. Math. , vol.49 , pp. 599-638
    • Lions, P.L.1    Perthame, B.2    Souganidis, P.E.3
  • 36
    • 84968518243 scopus 로고
    • A kinetic formulation of multidimensional conservation laws and related equations
    • [L-P-T1]. MR 1201239 (94d:35100)
    • [L-P-T1] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional conservation laws and related equations, Journal AMS 7 (1994), 169-191. MR 1201239 (94d:35100)
    • (1994) Journal AMS , vol.7 , pp. 169-191
    • Lions, P.L.1    Perthame, B.2    Tadmor, E.3
  • 37
    • 34249763254 scopus 로고
    • Kinetic formulation of isentropic gas dynamics and p-systems
    • [L-P-T2]. MR 1284790 (95c:76093)
    • [L-P-T2] P. L. Lions, B, Perthame and E. Tadmor, Kinetic formulation of isentropic gas dynamics and p-systems, Comm. Math. Physics 163 (1994), 415-431. MR 1284790 (95c:76093)
    • (1994) Comm. Math. Physics , vol.163 , pp. 415-431
    • Lions, P.L.1    Perthame, B.2    Tadmor, E.3
  • 38
    • 0033470637 scopus 로고    scopus 로고
    • A new entropy functional for a scalar conservation law
    • [Li-Y]. MR 1702712 (2000e:35139)
    • [Li-Y] T-P. Liu and T. Yang, A new entropy functional for a scalar conservation law, Comm. Pure and Appl. Math. 52 (1999), 1427-1442. MR 1702712 (2000e:35139)
    • (1999) Comm. Pure and Appl. Math. , vol.52 , pp. 1427-1442
    • Liu, T.-P.1    Yang, T.2
  • 40
    • 0032341805 scopus 로고    scopus 로고
    • On the equations describing a relaxation toward a statistical equilibrium state in two-dimensional perfect fluid dynamics
    • [M-R]. MR 1628271 (99d:76051)
    • [M-R] A. Mikelic and R. Robert, On the equations describing a relaxation toward a statistical equilibrium state in two-dimensional perfect fluid dynamics, SIAM J. Math. Analysis 29 (1998), 1238-1255. MR 1628271 (99d:76051)
    • (1998) SIAM J. Math. Analysis , vol.29 , pp. 1238-1255
    • Mikelic, A.1    Robert, R.2
  • 41
    • 0039565256 scopus 로고
    • Stable patterns in a viscous diffusion equation
    • [NC-P]. MR 1015926 (91f:35152)
    • [NC-P] A. Novick Cohen and R. Pego, Stable patterns in a viscous diffusion equation, Transactions AMS 324 (1991), 331-351. MR 1015926 (91f:35152)
    • (1991) Transactions AMS , vol.324 , pp. 331-351
    • Cohen, A.N.1    Pego, R.2
  • 42
    • 44749084234 scopus 로고
    • Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations
    • [O-S]. MR 0965860 (89h:80012)
    • [O-S] S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, Comput. Phys. 79 (1988) 12-49. MR 0965860 (89h:80012)
    • (1988) Comput. Phys. , vol.79 , pp. 12-49
    • Osher, S.1    Sethian, J.2
  • 45
    • 0000741448 scopus 로고
    • A kinetic equation with kinetic entropy functions for scalar conservation laws
    • [P-T]. MR 1099693 (92d:82095)
    • [P-T] B. Perthame and E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Physics 136 (1991), 501-517. MR 1099693 (92d:82095)
    • (1991) Comm. Math. Physics , vol.136 , pp. 501-517
    • Perthame, B.1    Tadmor, E.2
  • 46
    • 0013002645 scopus 로고
    • Analogy between thermodynamics and mechanics
    • [Pe]. MR 0532721 (80b:80014)
    • [Pe] M. A. Peterson, Analogy between thermodynamics and mechanics, American J. Physics 47 (1979), 488-490. MR 0532721 (80b:80014)
    • (1979) American J. Physics , vol.47 , pp. 488-490
    • Peterson, M.A.1
  • 48
    • 0012945863 scopus 로고
    • Passing to the limit with respect to the viscosity in an equation with variable parabolicity direction
    • [PI]
    • [PI] P. I. Plotnikov, Passing to the limit with respect to the viscosity in an equation with variable parabolicity direction, Differential Equations 30 (1994), 614-622.
    • (1994) Differential Equations , vol.30 , pp. 614-622
    • Plotnikov, P.I.1
  • 49
    • 0002117826 scopus 로고
    • Hausdorff measure and the Navier-Stokes equations
    • [S]. MR 0510154 (58:23176)
    • [S] V. Scheffer, Hausdorff measure and the Navier-Stokes equations, Comm. Math. Phys. 55 (1977), 97-112. MR 0510154 (58:23176)
    • (1977) Comm. Math. Phys. , vol.55 , pp. 97-112
    • Scheffer, V.1
  • 50
    • 0039271379 scopus 로고
    • Foundations of classical thermodynamics
    • [S1], Math. Department, U. of Chicago. MR 0672251 (58:32431)
    • [S1] J. Serrin, Foundations of Classical Thermodynamics, Lecture Notes, Math. Department, U. of Chicago, 1975. MR 0672251 (58:32431)
    • (1975) Lecture Notes
    • Serrin, J.1
  • 51
    • 0018685219 scopus 로고
    • Conceptual analysis of the classical second laws of thermodynamics
    • [S2]. MR 0574162 (81g:80002)
    • [S2] J. Serrin, Conceptual analysis of the classical second laws of thermodynamics, Arch. Rat. Mech. Analysis 70 (1979), 353-371. MR 0574162 (81g:80002)
    • (1979) Arch. Rat. Mech. Analysis , vol.70 , pp. 353-371
    • Serrin, J.1
  • 52
    • 0000104894 scopus 로고    scopus 로고
    • Statistical equilibrium and coherent states in two-dimensional turbulence
    • [Tu]. MR 1682816 (99k:76082)
    • [Tu] B. Turkington, Statistical equilibrium and coherent states in two-dimensional turbulence, Comm. in Pure and Applied Math. 52 (1999), 781-809. MR 1682816 (99k:76082)
    • (1999) Comm. in Pure and Applied Math. , vol.52 , pp. 781-809
    • Turkington, B.1
  • 56
    • 0042075546 scopus 로고
    • Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics
    • [W], Introduction to R. B. Israel, Princeton U. Press. MR 0517873 (80i:82002)
    • [W] A. Wightman, Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics, Introduction to R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton U. Press, 1979. MR 0517873 (80i:82002)
    • (1979) Convexity in the Theory of Lattice Gases
    • Wightman, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.