PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

A system of two conservation laws with flux conditions and small viscosity

Autorzy
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We construct explicit solutions of a system of two conservation laws with small viscosity in the quarter plane {(x, t): x > 0, t > 0}, with initial conditions at t = 0 and flux conditions at x = 0. We derive a formula for the limit as viscosity goes to zero which generally belongs to the space of locally bounded Borel measures. This limit satisfies the inviscid equation, in the sense of LeFloch [26]. We also treat more general initial and boundary datas and obtain solution in the algebra of generalized functions of Colombeau [9, 10].
Wydawca
Rocznik
Strony
247--267
Opis fizyczny
Bibliogr. 32 poz.
Twórcy
autor
Bibliografia
  • [1] Biagioni, H. A., Oberguggenberger, M., Generalized solutions to Burgers equation, J. Differential Equations 97 (1992), 263-287.
  • [2] Broadbridge, P., Knight, J. K., Rogers, C., Constant rate rain fall infiltration in a bounded profile: solution of a nonlinear model, Soil Sci. Soc. Amer. J. 52 (1988), 1526-1533.
  • [3] Broadbridge, P., Rogers, C., Exact solution for vertical drainage and redistribution in soils, J. Engrg. Math. 24 (1990), 25-43.
  • [4] Burger, R., Frid, H., Karlsen, K. H., On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition, J. Math. Anal. Appl. 326 (2007), 108-120.
  • [5] Chen, G. Q., Liu, H., Formation of delta-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM .J. Math. Anal. 34 (2003), 925-938.
  • [6] Clothier, B. E., Knight, J. K., White, I., Burgers equation: Application to field constant-flux infiltration, Soil Set.. 132 (1981), 225-261.
  • [7] Colombeau, J.F., New Generalized functions and Multiplication of distributions. North Holland, Amsterdam, 1984.
  • [8] Colombeau, J. F., Heibig, A., Generalized solutions to Cauchy problems. Monatsh. Math. 117 (1994), 33-49.
  • [9] Colombeau, J. F., Heibig, A., Oberguggenberger, M., The Cauchy problem in a space of generalized functions 1, C. R. Acad. Sci. Paris Ser. I Math. 317 (1993), 851-855.
  • [10] Colombeau, J. F., Heibig, A., Oberguggenberger. M., The Cauchy problem in a space. of generalized functions 2, C. R. Acad. Sci. Paris Ser. I Math. 319 (1994). 1179-1183.
  • [11] Conway, E. D., Hopf, E., Hamilton theory and generalized solution of the Hamiltion Jacobi equation, J. Math. Mech. 13 (1964), 939-986.
  • [12] Danilov, V. G., Shelkovich, V. M., Delta-shock wave type solution of hyperbolic systems of conservation laws, Quart. Appl. Math. 63(3) (2005), 401-427.
  • [13] Ercole, G., Delta-shock waves as self-similar viscosity limits, Quart. Appl. Math. 58(1) (2000), 177-199.
  • [14] Hopf, E., The partial differential equation ut + uux = vuxx, Comm. Pure Appl. Math. 3 (1950), 201-230.
  • [15] Huang, F., Existence and uniqueness of discontinuous solutions for a class of non-strictly hyperbolic systems, in "Advances in nonlinear partial differential equations and related areas" (Beijing 1997), World Sci. Publ., River Edge, NJ, 1999, 187-208.
  • [16] James, F., Convergence results for some conservation laws with a. reflux boundary condition and a relaxation term arising in chemical engineering, SIAM J. Math. Anal. 29(5) (1998), 1200-1223.
  • [17] Joseph, K. T., Burgers equation in the quarter plane, a formula, for the weak limit, Comm. Pure Appl. Math. 41 (1988), 133-149.
  • [18] Joseph, K. T., A Riemann problem whose viscosity solution contain ? -measures. Asymptotic Anal. 7 (1993), 105-120.
  • [19] Joseph, K. T., Generalized solutions to a Cauchy problem for a non- conservative hyperbolic system, J. Math. Anal. Appl. 207 (1997), 361-387.
  • [20] Joseph, K. T., Explicit generalized solutions to a system of conservation laws, Proc. Indian Acad. Sci. Math. Sci. 109 (1999), 401-409.
  • [21] Joseph, K. T., Explicit solutions for a system of first- order partial differential equations, Electron. J. Diff. Equ. Conf. 2008(157) (2008), 1-8.
  • [22] Joseph, K. T., Sachdev, P. L., Exact analysis of Burgers equation on the semi-line with flux conditions at the origin, Internat. J. Non-Linear Mech. 28 (1993). 627-639.
  • [23] Joseph, K. T., Vasudeva Murthy, A. S., Hopf-Cole transformation to some systems of partial differential equations, NoDEA Nonlinear Differential Equations Appl. 8 (2001), 173-193.
  • [24] Joseph, K. T., Veerappa Gowda, G. D., Explicit formula for the solution of convex conservation laws with boundary condition, Duke Math. J. 62 (1991), 401-416.
  • [25] Lax, P. D., Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537-566.
  • [26] LeFloch, P. G., An existence and uniqueness result for two non-strictly hyperbolic systems, in "Nonlinear evolution equations that change type". IMA Vol. Math. Appl. 27, Springer, New York, 1990, 126-139.
  • [27] Lions. P. L., Generalized solutions of Hamilton-Jacobi equation, Res. Notes Math. 69, Pitman, London, 1982.
  • [28] Philip, R. J., Theory of infiltration, Adv. Hydrosci. 5 (1969), 215-276.
  • [29] Shelkovich, V. M., The Riemann problem admitting ? - ?'-shocks, and vacuum states (the vanishing viscosity approach), Differential Equations 231 (2006), 459-500.
  • [30] Tan, D. Ch., Zhang, T., Zheng, Y. X. Delta-shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws, J. Differential Equations 112 (1994), 1-32.
  • [31] Volpert, A. I., The BV and quasi-linear equations, Math. USSR Sb. 2 (1967), 225-267.
  • [32] Weinan, E., Rykov, Yu. G., Sinai, Ya. G., Generalized varational principles. Global weak solutions and behaviour with random initial data for systems of conservation laws arising in adhension particle dynamics, Comm. Math. Phys. 177 (1996), 349-380.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-LOD6-0008-0031
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.