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Existence, uniqueness, and stability of generalized solutions of an initial-boundary value problem for a degenerating quasilinear parabolic equation. (English) Zbl 0891.35067

The authors continue a study of a spatially one-dimensional mathematical model for the settling and consolidation of a flocculated suspension [see, R. Bürger, W. Wendland, Entropy boundary and jump conditions in the theory of semidimentation with compression. Math. Methods Appl. Sci., to appear]. This model is formulated as the following initial-boundary value problem: \[ u_t+f_x(u,t)=a((u)u_x)_x,\quad (x,t)\in (0,1)\times (0,T); \]
\[ u(x,0)=u_0(x),\;x\in[0,1];\quad f_{bk}(u) - a(u)u_x|_{x=0} = 0,\quad t\in (0,T];\;u(1,t) = u_1(t),\;t\in(0,T]. \] Here, \(f\) is the flux density function such that \(f(u,t) = q(t)u + f_{bk}(u)\), \(u\) denotes the volumetric solid concentration, \(q(t)\) is the volume-averaged velocity of the suspension, \(f_{bk}\) the Kynch bath flux density function. The diffusion coefficient \(a\) is assumed to be a continuously differentiable function of \(u\) with \(a(u) = 0\) for \(u\leq \phi_c\) and \(u\geq 1\), and \(a(u)>0\) for \(\phi_c< u< 1\). This equation degenerates into first order hyperbolic type if the concentration is less than a critical value \(\phi_c\). In this paper, the existence, stability, and uniqueness of generalized solutions of this initial-boundary value problem are proved. The existence of generalized solutions is shown by the vanishing viscosity method.

MSC:

35K65 Degenerate parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35B45 A priori estimates in context of PDEs
35B35 Stability in context of PDEs
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[1] Bardos, C.; Le Roux, A. Y.; Nedelec, J. C., First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4, 1017-1034 (1979) · Zbl 0418.35024
[2] R. Bürger, Ein Anfangs-Randwertproblem einer quasilinearen entarteten parabolischen Gleichung in der Theorie der Sedimentation mit Kompression, University of Stuttgart, Germany, 1996; R. Bürger, Ein Anfangs-Randwertproblem einer quasilinearen entarteten parabolischen Gleichung in der Theorie der Sedimentation mit Kompression, University of Stuttgart, Germany, 1996
[3] R. Bürger, W. L. Wendland, Entropy boundary and jump conditions in the theory of sedimentation with compression, Math. Methods Appl. Sci.; R. Bürger, W. L. Wendland, Entropy boundary and jump conditions in the theory of sedimentation with compression, Math. Methods Appl. Sci.
[4] Concha, F.; Bustos, M. C.; Barrientos, A., Phenomenological theory of sedimentation, (Tory, E., Sedimentation of Small Particles in a Viscous Fluid (1996), Computational Mechanics: Computational Mechanics Southampton), 51-96
[5] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice Hall: Prentice Hall Englewood Cliffs · Zbl 0144.34903
[6] Gilding, B. H., A nonlinear degenerate parabolic equation, An. Scuola Norm. Sup. Pisa, 4, 393-432 (1977) · Zbl 0364.35027
[7] Kružkov, S. N., First order quasilinear equations in several independent variables, Math. USSR-Sb., 10, 217-243 (1970) · Zbl 0215.16203
[8] Kynch, G. J., A theory of sedimentation, Trans. Farad. Soc., 48, 166-176 (1952) · Zbl 0048.22902
[9] Ladyženskaja, O. A.; Ural’ceva, N. N., Boundary problems for linear and quasilinear equations, Amer. Math. Soc. Transl. Ser. 2, 47 (1965), Amer. Math. Soc: Amer. Math. Soc Providence, p. 217-299
[10] Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and Quasilinear Equations of Parabolic Type. Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, 23 (1968), Amer. Math. Soc: Amer. Math. Soc Providence
[11] Ole ı̆, O. A.; Kružkov, S. N., Quasi-linear second-order parabolic equations with many independent variables, Russian Math. Surveys, 16, 105-146 (1961) · Zbl 0112.32604
[12] Saks, S., Theory of the Integral. Theory of the Integral, Monografie Matematyczne, Druk. Yagiellonskiego, Warsaw (1937), G. E. Stechert & Co: G. E. Stechert & Co New York
[13] Ungarish, M., Hydrodynamics of Suspensions (1993), Springer-Verlag: Springer-Verlag New York/Berlin
[14] Vol’pert, A. I.; Hudjaev, S. I., Cauchy’s problem for degenerate second order quasilinear parabolic equations, Math. USSR-Sb., 7, 365-387 (1969) · Zbl 0191.11603
[15] Wloka, J., Partielle Differentialgleichungen (1982), Teubner-Verlag: Teubner-Verlag Stuttgart · Zbl 0482.35001
[16] Zhuoqun, Wu, Jun-yu, Wang, Some results on quasilinear degenerate parabolic equations of second order, in; Zhuoqun, Wu, Jun-yu, Wang, Some results on quasilinear degenerate parabolic equations of second order, in · Zbl 0522.35058
[17] Wu, Zhuoqun; Yin, Jingxue, Some properties of functions in\(BV_x\)and their applications to the uniqueness of solutions for degenerate quasilinear parabolic equations, Northeast. Math. J., 5, 395-422 (1989) · Zbl 0726.35071
[18] Zhao, Junning, Uniqueness of solutions of quasilinear degenerate parabolic equations, Northeast. Math. J., 1, 153-165 (1985) · Zbl 0604.35043
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