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Cauchy-Dirichlet problems for the porous medium equation

Dedicated to Professor Juan Luis Vázquez on the occasion of his 75th birthday

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  • We consider the porous medium equation subject to zero-Dirichlet conditions on a variety of two-dimensional domains, namely strips, slender domains and sectors, allowing us to capture a number of different classes of behaviours. Our focus is on intermediate-asymptotic descriptions, derived by formal arguments and validated against numerical computations. While our emphasis is on non-negative solutions to the slow-diffusion case, we also derive a number of results for sign-change solutions and for fast diffusion. Self-similar solutions of various kinds play a central role, alongside the identification of suitable conserved quantities. The characterisation of domains exhibiting infinite-time hole closure is a particular upshot and we highlight a number of open problems.

    Mathematics Subject Classification: Primary: 35K55, 35K20; Secondary: 35B40, 35R35.


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  • Figure 1.  Schematics of interfaces propagating to infinity. (a) Strip-like domain. (b) Slender domains, both expanding (to the right) and contracting (to the left). (c) Wedge-like: (i) acute angle; (ii) obtuse angle; only the details of the large-time behaviour differ between acute and obtuse angles – see Section 5. In (a) and (c) the shape of the domain towards the left has negligible bearing on the large-time behaviour of the spreading to the right; the converse applies to the spreading to the left in Figure 2

    Figure 2.  Schematics of interfaces propagating into a singular point on the boundary. (a) Cusp (locally slender domain). (b) Corner

    Figure 3.  Three classes of similarity solutions for the porous medium equation on different domains: (a) on a semi-infinite strip $ D = \{x\ge 0, -1\le y\le 1\} $, (b) on the first quadrant $ D = \{x\ge 0, y\ge 0\} $, and (c) on the half-plane $ D = \{x\ge 0\} $. The profiles were computed for (1) with $ m = 1 $. Arrows indicate the direction of spreading further into the domain

    Figure 4.  Comparing the long-time dynamics of the similarity solutions from Figure 3, on (a) the semi-infinite strip (red solid curves), (b) first quadrant (blue dashed curves), (c) half-plane (black dot-dashed curves): (left) all three approach power-law decays $ O(t^{-\alpha}) $ for the maximum of the solution, $ u_{\max}(t) = \max_D u(x,y,t) $, but (right) while solutions (b, c) approach power-law spreading of the region of support $ O(t^\beta) $ for $ x_{\mathrm{edge}}(t) = \mathrm{argmax}_D \{ u(x,y,t)>0\} $, the edge of support of solution (a) advances logarithmically

    Figure 5.  Computed solutions on the semi-infinite strip for $ m = 1 $. (top) 3-D cut-away view of $ u(x,y,t) $ (axes not to scale), cross-sections being indicated by the blue and black curves. (left) Time profiles of the scaled centreline profile, $ u(x,0,t)/u_{\max}(t) $ at increasing times, (right) Scaled cross-section profiles mid-way through the computed PDE solution, $ u(x_{\mathrm{edge}}(t)/2,y,t)/u_{\max}(t) $, at corresponding times, consistent with the 1-D zeroth-kind separable solution $ F(y) $ of (29)

    Figure 6.  Numerically computed PDE solutions at large times for (1) with $ m = 1 $ for: (left) the quarter plane problem, $ \phi = \pi/4 $, (middle) the 3-quarter plane problem, $ \phi = 3\pi/4 $, and (right) the slit-plane problem, $ \phi = \pi $

    Figure 7.  Cut-away view of the numerically computed half-plane similarity solution for $ m = 2 $. It is shown in terms of the pressure $ V(\xi,\eta) = U(\xi,\eta)^m $. The solution is symmetric about the line $ \eta = 0 $. The centreline profile (red curve), $ V(\xi,0) $, and the free boundary (black curve), $ V(\xi,\eta) = 0 $, are shown in more detail in Figure 13

    Figure 8.  (left) Profiles with $ m = 1 $ for similarity solutions (the non-negative solution (black), and solutions with $ n = 1, 2,3 $ sign changes (red, blue, and green curves respectively)). Solutions are normalised to satisfy condition (58). (right) The corresponding $ k = \alpha/\beta $ ratios over a range of $ m $, ending with the first-kind solution values $ k_n = 2(n+1) $ at $ m = 0 $

    Figure 9.  Numerical simulations of (9) with $ m = 1 $ in the upper half plane, showing contours of the self-similar solution at a typical large time, starting from different initial data: (a) the non-negative half-plane solution, (b) a pair of oppositely signed quarter plane solutions (contours corresponding to positive solution values are shown as solid red curves, while contours for negative solution values are shown as blue dashed curves), (c) a second-kind similarity solution with a single non-trivial nodal curve, (d) three sectorial solutions with angle $ \phi = \pi/6 $ with alternating signs

    Figure 10.  Numerical simulations of (9) with $ m = 1 $ in the quarter plane given by the first quadrant, showing contours of the self-similar solution at a typical large time, starting from different initial data: (left) a pair of oppositely signed $ \phi = \pi/8 $ sector first-kind similarity solutions, (middle) a second-kind similarity solution with a single non-trivial nodal curve, (right) a pair of oppositely signed $ \phi = \pi/8 $ second-kind similarity solutions

    Figure 11.  Diagram of $ (m,\phi) $-space illustrating the different regions corresponding to the two possible leading order terms at the vertex of sectors. The separating curve is given by critical sector angle (60)

    Figure 12.  (left) Schematic showing definitions of $ r_{\min}(t) $ and $ r_{\max}(t) $ for the region of support of the spreading solution in a sector with $ \phi = \pi/8 $, corresponding to the left-hand panel of Figure 10. (right) Simulation results for $ r_{\min}, r_{\max} $ for $ m = 3/4 $ (blue dashed curves) and $ m = 2 $ (red solid curves) along with the asymptotic predictions (black dotted curves): infinite time spreading for $ m>m_c $ and finite-time corner-filling for $ m<m_c $, where $ m_c = 1 $ for $ \phi = \pi/8 $. The three infinite-time predictions are those obtained from the first-kind solutions; the finite-time one (for $ r_{\min} $ with $ m<m_c $) is a fit to an a priori unknown power law, $ r\propto (t_c-t)^{\gamma} $

    Figure 13.  (left) Centreline profile $ V(\xi,0) $ for the $ m = 2 $ half-plane similarity solution (solid red) compared with the two-term local expansions at $ \xi = 0 $ (101) (dash-dotted black) and at the tip $ \xi = \xi_0 $ (103) (dashed blue), (middle) Half-view of the free-boundary (solid black) compared with local quadratic form at the tip (104) (dash-dotted blue) and also fitted to a shifted ellipse (105) (dashed red), and (right) The flux $ J(\eta) $ (see (100)) through the fixed boundary $ \xi = 0 $

  • [1] S. B. Angenent and D. G Aronson, Optimal asymptotics for solutions to the initial value problem for the porous medium equation, Nonlinear Problems in Applied Mathematics, (1996), 10-19.
    [2] S. B. AngenentD. G. AronsonS. I. Betelu and J. S. Lowengrub, Focusing of an elongated hole in porous medium flow, Phys. D, 151 (2001), 228-252.  doi: 10.1016/S0167-2789(01)00150-6.
    [3] D. G. Aronson and L. A. Peletier, Large time behaviour of solutions of the porous medium equation in bounded domains, J. Differential Equations, 39 (1981), 378-412.  doi: 10.1016/0022-0396(81)90065-6.
    [4] G. I. Barenblatt, On some unsteady motions of a fluid and a gas in a porous medium, Prikl. Mat. Makh., 16 (1952), 67-78. 
    [5] G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics, Consultants Bureau [Plenum], New York-London, 1979.
    [6] G. I. BarenblattScaling, Self-Similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics, Cambridge University Press, 1996.  doi: 10.1017/CBO9781107050242.
    [7] J. Bear, Dynamics of Fluids in Porous Media, Dover, 1988. doi: 10.1097/00010694-197508000-00022.
    [8] S. I. Betelú and J. R. King, Explicit solutions of a two-dimensional fourth-order nonlinear diffusion equation, Math. Comput. Modelling, 37 (2003), 395-403.  doi: 10.1016/S0895-7177(03)00015-3.
    [9] A. Friedman and S. Kamin, The asymptotic behavior of gas in an $n$-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.  doi: 10.2307/1999846.
    [10] J. Gilchrist, Flux diffusion and the porous medium equation, Phys. C, 291 (1997), 132-142.  doi: 10.1016/S0921-4534(97)01685-7.
    [11] B. H. Gilding and J. Goncerzewicz, The porous media equation in an infinite cylinder, between two infinite parallel plates, and like spatial domains, Interfaces and Free Boundaries, 18 (2016), 45-73.  doi: 10.4171/IFB/356.
    [12] J. Hulshof, Similarity solutions of the porous medium equation with sign changes, J. Math. Anal. Appl., 157 (1991), 75-111.  doi: 10.1016/0022-247X(91)90138-P.
    [13] J. HulshofJ. R. King and M. Bowen, Intermediate asymptotics of the porous medium equation with sign changes, Adv. Differential Equations, 6 (2001), 1115-1152. 
    [14] J. Hulshof and J. L. Vázquez, The dipole solution for the porous medium equation in several space dimensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20 (1993), 193-217, http://www.numdam.org/item/?id=ASNSP_1993_4_20_2_193_0.
    [15] S. Kamin and J. L. Vazquez, Asymptotic behaviour of solutions of the porous medium equation with changing sign, SIAM J. Math. Anal., 22 (1991), 34-45.  doi: 10.1137/0522003.
    [16] W. L. Kath and D. S. Cohen, Waiting-time behavior in a nonlinear diffusion equation, Stud. Appl. Math., 67 (1982), 79-105.  doi: 10.1002/sapm198267279.
    [17] J. Kevorkian and J. D. Cole., Perturbation Methods in Applied Mathematics, volume 34 of Applied Mathematical Sciences., Springer-Verlag, New York-Berlin, 1981.
    [18] J. R. King, Integral results for nonlinear diffusion equations, J. Engrg. Math., 25 (1991), 191-205.  doi: 10.1007/BF00042853.
    [19] J. R. King, Exact multidimensional solutions to some nonlinear diffusion equations, Quart. J. Mech. Appl. Math., 46 (1993), 419-436.  doi: 10.1093/qjmam/46.3.419.
    [20] J. R. King, Self-similar behaviour for the equation of fast nonlinear diffusion, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 343 (1993), 337-375.  doi: 10.1098/rsta.1993.0052.
    [21] J. R. King, Asymptotic analysis of extinction behaviour in fast nonlinear diffusion, J. Engrg. Math., 66 (2010), 65-86.  doi: 10.1007/s10665-009-9329-4.
    [22] J. R. King, Unpublished notes, 2021.
    [23] R. C. Kloosterziel, On the large-time asymptotics of the diffusion equation on infinite domains, J. Engrg. Math., 24 (1990), 213-236.  doi: 10.1007/BF00058467.
    [24] A. A. LaceyJ. R. Ockendon and A. B. Tayler, "Waiting-time" solutions of a nonlinear diffusion equation, SIAM J. Appl. Math., 42 (1982), 1252-1264.  doi: 10.1137/0142087.
    [25] W. I. Newman, Some exact solutions to a nonlinear diffusion problem in population genetics and combustion, J. Theoret. Biol., 85 (1980), 325-334.  doi: 10.1016/0022-5193(80)90024-7.
    [26] W. I. Newman, The long-time behavior of the solution to a nonlinear diffusion problem in population genetics and combustion, J. Theoret. Biol., 104 (1983), 473-484.  doi: 10.1016/0022-5193(83)90240-0.
    [27] R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math., 12 (1959), 407-409.  doi: 10.1093/qjmam/12.4.407.
    [28] A. Rodríguez and J. L. Vázquez, Non-uniqueness of solutions of nonlinear heat equations of fast diffusion type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 173-200.  doi: 10.1016/s0294-1449(16)30163-9.
    [29] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, volume 33 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780199202973.001.0001.
    [30] J. L. VázquezThe Porous Medium Equation, Oxford University Press, Oxford, 2007. 
    [31] J. L. Vazquez, Porous medium flow in a tube: Traveling waves and KPP behavior, Commun. Contemp. Math., 9 (2007), 731-751.  doi: 10.1142/S0219199707002587.
    [32] T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Studies in Applied Mathematics, 100 (1998), 153-193.  doi: 10.1111/1467-9590.00074.
    [33] T. P. Witelski and M. Bowen, ADI schemes for higher-order nonlinear diffusion equations, Appl. Numer. Math., 45 (2003), 331-351.  doi: 10.1016/S0168-9274(02)00194-0.
    [34] Y. B. Zel'dovič and G. I. Barenblatt, Asymptotic properties of self-preserving solutions of equations of unsteady motion of gas through porous media, In Doklady Akademii Nauk, volume 118,671-674. Russian Academy of Sciences, 1958.
    [35] Ya. B. Zeldovich and G. I. Barenblatt, On the dipole-type solution in the problems of a polytropic gas flow in porous medium, Prikl. Mat. Mekh, 21 (1957), 718-720. 
    [36] Ya. B. Zel'dovič and A. S. Kompaneets, On the theory of propagation of heat with the heat conductivity depending upon the temperature, In Collection in Honor of the Seventieth Birthday of Academician A. F. Ioffe, 61-71. Izdat. Akad. Nauk SSSR, Moscow, 1950.
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