=1 =amstex R L^1() L^&infin#infty;() BV() 0  ·  BRADLEY J. LUCIER""Department of Mathematics, Purdue University, West Lafayette, Indiana 47907. The work of the first author was not supported by the Wolf Foundation. and DOUGLAS N. ARNOLD""Department of Mathematics, University of Maryland, College Park, Maryland 20742.

Abstract:

This nonsense paper exemplifies the IMA preprint style file for AmS-TeX  imappt.sty. The IMA preprint style file, which is used by the Institute for Mathematics and its Applications for its preprints, is offered as a general purpose preprint style for mathematical papers. It is a modification of the amsppt style of Michael Spivak. This paper also serves to illustrate many of the amstex macros as used with the imappt style file.

1. Introduction We are concerned with numerical approximations to the so-called porous-medium equation [#!5!#],

$$\alignedat2 &u_t=\phi(u)_{xx},&&\qquad x\in\BbbR,\quad t>0,\quad... ...=u^m,\quad m>1, \\ &u(x,0)=u_0(x),&&\qquad x\in\BbbR. \endalignedat \tag 1.1$$

We assume that the initial data u₀(x) has bounded support, that 0≤u₀≤M, and that φ(u₀)_x∈$\bvr$. It is well known that a unique solution u(x, t) of (1.1) exists, and that u satisfies

0≤u≤M and $$\TV$$φ(u( ⋅ , t))_x≤$$\TV$$φ(u₀)_x.$$\tag$$1.2

If the data has slightly more regularity, then this too is satisfied by the solution. Specifically, if m is no greater than two and u₀ is Lipschitz continuous, then u( ⋅ , t) is also Lipschitz; if m is greater than two and (u₀^m-1)_x∈$\linfr$, then (u( ⋅ , t)^m-1)_x∈$\linfr$ (see [3]). (This will follow from results presented here, also.) We also use the fact that the solution u is Hölder continuous in t.

2. $\linfr$ error bounds After a simple definition, we state a theorem that expresses the error of approximations u^h in terms of the weak truncation error E. Definition 2.1A definition is the same as a theorem set in roman type. Theorem 2.1 Let {u^h} be a family of approximate solutions satisfying the following conditions for 0≤t≤T: For all x∈$\BbbR$ and positive t, 0≤u^h(x, t)≤M; Both u and u^h are Hölder–α in x for some α∈(0, 1∧1/(m - 1)); u^h is right continuous in t; and u^h is Hölder continuous in t on strips $\BbbR$×(tⁿ, tⁿ⁺¹), with the set {tⁿ} having no limit points; and There exists a positive function ω(h, ε) such that: whenever {w^ε}_{0 < ε≤ε0} is a family of functions in $\bold$X for which "(a)" there is a sequence of positive numbers ε tending to zero, such that for these values of ε, | w^ε|_∞≤1/ε, "(b)" for all positive ε, | w_x^ε($\sdot$, t)|_$\loner$≤1/ε², and "(c)" for all ε > 0,

sup$$\Sb$$x∈$$\BbbR$$
0≤t₁, t₂≤T$$\Sb$$$${\dfrac{{\vert w^\epsilon(x,t_2)-w^\epsilon(x,t_1)\vert}}{{\vert t_2-t_1\vert^p}}}$$≤1/ε²,

where p is some number not exceeding 1, then | E(u^h, w^ε, T)|≤ω(h, ε). Then, there is a constant C = C(m, M, T) such that

| u - u^h|_{∞,$\BbbR$×[0, T]}≤C$$\left[\vphantom{ \sup \left \vert\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx\right\vert+ \omega(h,\epsilon)+\epsilon^\alpha}\right.$$sup$$\left\vert\vphantom{\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx}\right.$$$$\int_{\BbbR}^{}$$(u₀(x) - u^h(x, 0))w(x, 0) dx$$\left.\vphantom{\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx}\right\vert$$ + ω(h, ε) + ε^α$$\left.\vphantom{ \sup \left \vert\int_\BbbR(u_0(x)-u^h(x,0)) w(x,0) \,dx\right\vert+ \omega(h,\epsilon)+\epsilon^\alpha}\right]$$,$$\tag$$2.1

where the supremum is taken over all w∈$\bold$X. ProofLet z be in $\bold$X. Because E(u,$\sdot$,$\sdot$)≡ 0, Equation (1.5) implies that

$$\int_{\BbbR}^{}$$Δuz|^T₀dx = $$\int_{0}^{T}$$$$\int_{\BbbR}^{}$$Δu(z_t + φ[u, u^h]z_xx) dx dt - E(u^h, z, t),$$\tag$$2.2

where Δu = u - u^h and

φ[u, u^h] = $${\dfrac{{\phi(u)-\phi(u^h)}}{{u-u^h}}}$$.

Extend φ[u, u^h](⋅, t) = φ[u, u^h](⋅, 0) for negative t, and φ[u, u^h](⋅, t) = φ[u, u^h](⋅, T) for t > T.Fix a point x₀ and a number ε > 0. Let j_ε be a smooth function of x with integral 1 and support in [- ε, ε], and let J_δ be a smooth function of x and t with integral 1 and support in [- δ, δ]×[- δ, δ]; δ and ε are positive numbers to be specified later. We choose z = z^εδ to satisfy

$$\aligned &z_t+(\delta+J_\delta*\phi[u,u^h])z_{xx}=0,\qquad x\in\BbbR,\;0\leq t\leq T, \\ &z(x,T)=j_\epsilon(x-x_0). \endaligned \tag 2.3$$

The conclusion of the theorem now follows from (2.1) and the fact that

| j_ε*Δu(x₀, t) - Δu(x₀, t)|≤Cε^α,

which follows from Assumption 2.$\qedsymbol$ #_no#> 1 K. Hollig and M. Pilant Regularity of the free boundary for the porous medium equation MRC Tech. Rep. 2742 #_no#> 2 J. Jerome Approximation of Nonlinear Evolution Systems Academic Press New York 1983 #_no#> 3 R. J. LeVeque Convergence of a large time step generalization of Godunov's method for conservation laws Comm. Pure Appl. Math. 37 1984 463–478 #_no#> 4 A large time step generalization of Godunov's method for systems of conservation laws #_no#> 5 B. J. Lucier On nonlocal monotone difference methods for scalar conservation laws Math. Comp. 47 1986 19–36 ;''ARRAY(0x1e0a0158)