Abstract
We approximate the unique entropy solutions to general multidimensional degenerate parabolic equations with BV continuous flux and continuous nondecreasing diffusion function (including scalar conservation laws with BV continuous flux) in the periodic case. The approximation procedure reduces, by means of specific formulas, a system of PDEs to a family of systems of the same number of ODEs in the Banach space \(L^\infty \), whose solutions constitute a weak asymptotic solution of the original system of PDEs. We establish well posedness, monotonicity and \(L^1\)-stability. We prove that the sequence of approximate solutions is strongly \(L^1\)-precompact and that it converges to an entropy solution of the original equation in the sense of Carrillo. This result contributes to justify the use of this original method for the Cauchy problem to standard multidimensional systems of fluid dynamics for which a uniqueness result is lacking.
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References
Albeverio, S., Danilov, V.G.: Construction of global in time solutions to Kolmogorov–Feller pseudodifferential equations with a small parameter using characteristics. Math. Nach. 285(4), 426–439 (2012)
Albeverio, S., Shelkovich, V.M.: On delta shock problem. In: Rozanova, O. (ed.) Analytical Approaches to Multidimensional Balance Laws, pp. 45–88. Nova Science Publishers, New York (2005)
Andreianov, B., Maliki, M.: A note on uniqueness of entropy solutions to degenerate parabolic equations in \({\mathbb{R}}^N\). NoDEA: Nonlinear Differ. Equ. Appl. 17(1), 109–118 (2010)
Bear, J., Cheng, A.H.D.: Modeling Groundwater Flow and Contaminant Transport. Theory and Applications of Transport in Porous Media, vol. 23. Springer, Dordrecht (2011)
Bebernes, J., Eberly, D.: Mathematical Problems from Combustion Theory. Applied Mathematical Sciences, vol. 83. Springer, New York (2013)
Carrillo, J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147, 269–361 (1999)
Colombeau, M.: Weak asymptotic methods for 3-D self-gravitating pressureless fluids; application to the creation and evolution of solar systems from the fully nonlinear Euler–Poisson equations. J. Math. Phys. 56, 061506 (2015)
Colombeau, M.: Approximate solutions to the initial value problem for some compressible flows. Zeitschrift fur Angewandte Mathematik und Physik 66(5), 2575–2599 (2015)
Colombeau, M.: Asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form. J. Differ. Equ. 260(1), 197–217 (2016)
Danilov, V.G., Omel’yanov, G.A., Shelkovich, V.M.: Weak asymptotic method and interaction of nonlinear waves. AMS Trans. 208, 33–164 (2003)
Danilov, V.G., Mitrovic, D.: Delta shock wave formation in the case of triangular hyperbolic system of conservation laws. J. Differ. Equ. 245, 3704–3734 (2008)
Danilov, V.G., Shelkovich, V.M.: Dynamics of propagation and interaction of \( \delta \) shock waves in conservation law systems. J. Differ. Equ. 211, 333–381 (2005)
Danilov, V.G., Shelkovich, V.M.: Delta-shock wave type solution of hyperbolic systems of conservation laws. Q. Appl. Math. 63, 401–427 (2005)
Gerritsen, M., Durlofsky, L.J.: Modeling of fluid flow in oil reservoirs. Ann. Rev. Fluid Mech. 37, 211–238 (2005)
Godunov, A.N.: Peano’s theorem in Banach spaces. Funktsional. Anal. i Prilozhen. 9(1), 59–60 (1975). (Russian)
Joseph, K.T., Sahoo, M.R.: Boundary Riemann problems for the one dimensional adhesion model. Can. Appl. Math. Q. 19, 19–41 (2011)
Joseph, K.T., Choudury, A.P., Sahoo, M.R.: Spherical symmetric solutions of multidimensional zero pressure gas dynamics system. J. Hyperbolic Differ. Equ. 11(2), 269–294 (2014)
Joseph, K.T., Sahoo, M.R.: Vanishing viscosity approach to a system of conservation laws admitting delta waves. Commun. Pure Appl. Anal. 12(6), 2091–2118 (2013)
Karlsen, K.H., Lie, K.A.: An unconditionally stable splitting scheme for a class of nonlinear parabolic equations. IMA J. Numer. Anal. 19(4), 609–635 (1999)
Kruzhkov, S.N.: First order quasilinear equations in several independent variables. Mat. Sb. 81, 228–255 (1970). [(Russian). English Translation in Math USSR Sb. 10, 217–243 (1970)]
Kruzhkov, S.N., Panov, E.Yu.: First-order conservative quasilinear laws with an infinite domain of dependence on the initial data. Dokl. Akad. Nauk SSSR 314, 79–84 (1990). [(Russian). English Translation in Soviet Math. Dokl. 42, 316–321 (1991)]
Kruzhkov, S.N., Panov, EYu.: Osgood’s type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order. Ann. Univ. Ferrara Sez. VII (N.S.) 40, 31–54 (1994)
Kmit, I., Kunzinger, M., Steinbauer, R.: Generalized solutions of the Vlasov–Poisson system with singular data. J. Math. Anal. Appl. 340(1), 575–587 (2008)
Kunzinger, M., Rein, G., Steinbauer, R., Teschl, G.: Global weak solution of the relativistic Vlasov–Klein Gordon system. Commun. Math. Phys. 238(1–2), 367–378 (2003)
Maliki, M., Touré, H.: Uniqueness of entropy solutions for nonlinear degenerate parabolic problem. J. Evol. Equ. 3(4), 603–622 (2003)
Panov, EYu., Shelkovich, V.M.: \(\delta \)’-shock waves as a new type of solutions to systems of conservation laws. J. Differ. Equ. 228, 49–86 (2006)
Panov, EYu.: On the decay property for periodic renormalized solutions to scalar conservation laws. J. Differ. Equ. 260(3), 2704–2728 (2016)
Sahoo, M.R.: Generalized solutions to a system of conservation laws which is not strictly hyperbolic. J. Math. Anal. Appl. 432(1), 214–232 (2015)
Sahoo, M.R.: Weak asymptotic solutions for a nonstrictly hyperbolic system of conservation laws. Electron. J. Differ. Equ. 2016(94), 1–14 (2016)
Shelkovich, V.M.: \(\delta \)- and \(\delta ^{\prime }\)-shock wave types of singular solutions of systems of conservation laws and transport and concentration processes. Russ. Math. Surv. 63(3), 405–601 (2008)
Shelkovich, V.M.: The Riemann problem admitting \(\delta \)-, \(\delta ^{\prime }\)-shocks and vacuum states; the vanishing viscosity approach. J. Differ. Equ. 231, 459–500 (2006)
Tory, E.M., Karlsen, K.H., Bürger, R., Berres, S.: Strongly degenerate parabolic–hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64(1), 41–80 (2003)
Vazquez, J .L.: The Porous Medium Equation: Mathematical Theory. Oxford Mathematical Monographs. Clarendon Press, Oxford (2006)
Whitham, G.B.: Linear and Nonlinear Waves. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts, vol. 42. Wiley, Hoboken (2011). (reprint)
Zeidler, E.: Nonlinear Functional analysis and Its Applications. II/A: Linear Monotone Operators, vol. 467. Springer, New York (1989)
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Abreu, E., Colombeau, M. & Panov, E.Y. Approximation of entropy solutions to degenerate nonlinear parabolic equations. Z. Angew. Math. Phys. 68, 133 (2017). https://doi.org/10.1007/s00033-017-0877-6
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DOI: https://doi.org/10.1007/s00033-017-0877-6
Keywords
- Partial differential equations
- Degenerate parabolic equations
- Entropy solutions
- Approximate solutions
- Stability