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.
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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 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 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 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 $
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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
Schematics of interfaces propagating into a singular point on the boundary. (a) Cusp (locally slender domain). (b) Corner
Three classes of similarity solutions for the porous medium equation on different domains: (a) on a semi-infinite strip
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
Computed solutions on the semi-infinite strip for
Numerically computed PDE solutions at large times for (1) with
Cut-away view of the numerically computed half-plane similarity solution for
(left) Profiles with
Numerical simulations of (9) with
Numerical simulations of (9) with
Diagram of
(left) Schematic showing definitions of
(left) Centreline profile