KAZAMA, MASAKI

OMATA, SEIRO

SVADLENKA, KAREL

Kyoto University Research Information Repository

Title Discrete Morse flow for nonlocal problems (Problems in theCalculus of Variations and Related Topics)Author(s)OMATA, SEIRO; KAZAMA, MASAKI; SVADLENKA,KARELCitation数理解析研究所講究録 (2009), 1628: 42-47Issue Date2009-02URL http://hdl.handle.net/2433/140345RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityDiscrete Morse flow for nonlocal problemsSEIRO OMATA MASAKI KAZAMA KAREL \v{S}VADLENKAGraduate School of Natural Science and Technology, Kanazawa UniversityKakuma-machi, Kanazawa, 920-1192 Japan(omata@kenroku.kanazawa-u.ac.jp)Abstract. The analysis of evolutionary nonlocal problems represented by volume-constmined motion is discussed. The variational approach called the discrete Morse semi-flow proves to be a suitable tool.In this contribution, we consider the constrained film motion described by the followingequations:$\rho u_{u}(t,x)=\gamma\Delta u(t, x)+\lambda(t)$ for $(t, x)\in(O, T)\cross\Omega\cap\{u>0\}$ , (1)$\frac{\gamma}{2}|\nabla u|^{2}-\frac{\rho}{2}u_{t}^{2}=Q^{2}$ on $(0, T)\cross\Omega\cap\partial\{u>0\}$ , (2)$u(0, x)=u_{0}(x)$ , $u_{t}(0, x)=v_{0}(x)$ in $\Omega$ . (3)Here $u$ is a scalar function $(0, T)\cross\Omega\mapsto R$ expressing the shape of film, $\Omega$ is a domain in$R^{m}$ and $\rho,$ $\gamma$ and $Q$ are given positive constants. The parameter $Q$ describes the adhesiveor surface tension properties of the materials. The value of $u$ is set to zero outside of$\{u>0\}$ which is the set $\{(t, x)\in[0, T]\cross\overline{\Omega};u(t, x)>0\}$ . Finally, $\lambda$ is a Lagrangemultiplier originating in the requirement of volume preservation$\int_{\Omega}u(t, x)dx=V$ $\forall t\in[0, T]$ , (4)and is defined by$\lambda=\frac{1}{V}\int_{\Omega}[\rho u_{tt}u+\gamma|\nabla u|^{2}]dx$ . (5)Typical phenomena described by these equations without the volume constraint arevibration and peeling of a film. The volume-preserving equation can model motion ofbubbles and droplets attached to surfaces (i.e., obstacles).These equations are derived by the following consideration. First, we suppose thatthe potential energy for the shape of the film is described by the formula$\int_{\Omega}(\frac{\gamma}{2}|\nabla u|^{2}+Q^{2}\chi_{u>0})dx$,where $\chi_{u>0}$ is the characteristic function of the set $\{u>0\}$ . We define the action intergralof the film as$J(u)= \int_{0}^{T}\int_{\Omega}(\frac{\gamma}{2}|\nabla u|^{2}+Q^{2}\chi_{u>0}-\frac{\rho}{2}u_{t}^{2}\chi_{u>0})dxdt$. (6)1628 2009 42-47 42Then we can derive the film equation (1) by calculating the first variation using volume-preserving perturbations$\frac{dJ(\frac{u+\delta\zeta}{1+(\delta/V)\int\zeta d\delta})}{}|_{\delta=0}=0$for any $\zeta\in C_{0}^{\infty}((0, T)\cross\Omega\cap\{u>0\})$ .We also derive the free boundary condition (2) by the following calculation. Let$\eta\in C_{0}^{\infty}((0, T)\cross\Omega, R\cross R^{m}),$ $z=(t, x)\in(0, T)\cross\Omega$ , define $\tau_{\delta}(z)=z+\delta\eta$ and choose$K_{\delta}$ in such a way that the perturbation $K_{\delta}u(\tau_{\delta}^{-1}(z))$ preserves volume. By carrying outthe inner variation$\frac{dJ(K_{\delta}u(\tau_{\delta}^{-1}(z)))}{d\delta}|_{\delta=0}=0$ ,we get the free boundary condition (2).We study these equations by the discrete Morse flow approach introduced first in [1].The discrete Morse flow is a method that solves time-dependent problems by discretiz-ing time and defining a sequence of minimization problems approximating the originalproblem. The corresponding minimizers are then interpolated with respect to time anddiscretization parameter is sent to zero. The advantage of this method over other ap-proaches lies mainly in its constructivity and in the absence of restrictive assumptions,such as convexity.We shall explain the idea on the example of the wave equation ([5]). We consider thefollowing problem:$u_{tt}(t, x)$ $=$ $\Delta u(t, x)$ in $Q_{T}=(0, T)\cross\Omega$ , (7)$u(t, x)$ $=0$ on $(0, T)\cross\partial\Omega$ , (8)$u(O, x)$ $=u_{0}(x)$ , $u_{t}(0, x)=v_{0}(x)$ in $\Omega$ . (9)First, we fix a natural number $N>0$ , determine the time step $h=T/N$ and put$u_{1}(x)=u_{0}(x)+hv_{0}(x)$ . Function $u_{0}$ corresponds to the approximate solution at timelevel $t=0$ , while function $u_{1}$ is the approximate solution at time level $t=h$ . We definethe approximate solution $u_{n}$ on further time levels $t=nh$ for $n=2,3,$ $\ldots,$ $N$ , to be aminimizer of the following functional in $H_{0}^{1}(\Omega)$ :$J_{n}(u)= \int_{\Omega}\frac{|u-2u_{n-1}+u_{n-2}|^{2}}{2h^{2}}dx+\frac{1}{2}\int_{\Omega}|\nabla u|^{2}dx$ . (10)We observe that the second term of the functional is lower-semicontinuous with respectto sequentially weak convergence in $H^{1}(\Omega)$ and the first term is continuous in $L^{2}(\Omega)$ . Theexistence of minimizers then follows immediately from the fact that the functionals arebounded from below. This is a crucial advantage over the continuous version of thisfunctional, the Lagrangian of the type (6).43As the next step, we define the approximate solutions $\overline{u}^{h}$ and $u^{h}$ through interpolationof the minimizers $\{u_{n}\}_{n=0}^{N}$ in time:$\overline{u}^{h}(t, x)=\{\begin{array}{l}u_{0}(x), t=0u_{n}(x), t\in((n-1)h, nh], n=1, \ldots, N,\end{array}$ (11)$u^{h}(t, x)=\{\begin{array}{l}u_{0}(x), t=0\frac{t-(n-1)h}{h}u_{n}(x)+\frac{nh-t}{h}u_{n-1}(x), t\in((n-1)h, nh], n=1, \ldots, N.\end{array}$Since $u_{n}$ is a minimizer of $J_{n}$ , the first variation of $J_{n}$ at $u_{n}$ vanishes. Thus, for any$\varphi\in H_{0}^{1}(\Omega)$ we have$\int_{h}^{T}\int_{\Omega}[\frac{u_{t}^{h}(t)-u_{t}^{h}(t-h)}{h}\varphi+\nabla\overline{u}^{h}\nabla\varphi]dxdt=0$ $\forall\varphi\in L^{2}(0, T;H_{0}^{1}(\Omega))$ . (12)Now, we take the time step to zero. To be able to do so, some estimate on theapproximate solutions is needed. We obtain the following energy estimate:$||u_{t}^{h}(t)\Vert_{L^{2}(\Omega)}+\Vert\nabla\overline{u}^{h}(t)\Vert_{L^{2}(\Omega)}\leq C_{E}$ for a.e. $t\in(O, T)$ , (13)where constant $C_{E}$ depends on $H^{1}$-norms of the initial data but is independent of $h$ .Thanks to estimate (13), we can extract a subsequence converging weakly to a function$u\in H^{1}(Q_{T})$ and pass to limit in (12) as $harrow 0+$ to conclude that$\int_{0}^{T}\int_{\Omega}(-u_{t}\varphi_{t}+\nabla u\nabla\varphi)dxdt-\int_{\Omega}v_{0}\varphi(0, x)dx=0$ $\forall\varphi\in C_{0}^{\infty}([0, T)\cross\Omega)$ . (14)Moreover, it can be shown that $u$ satisfies boundary condition (8) and remaining initialcondition (9) in the sense of traces. Therefore, we have proved by the discrete Morse flowmethod that there exists a weak solution $u\in H^{1}(Q_{T})$ to problem (7) $-(9)$ .The method of discrete Morse flow can be naturally applied to nonlocal problemsand extends even to free-boundary problems. The advantage of our approach regardingglobally constrained problems lies in the fact that a semi-discretization of time allows usto use results from elliptic theory. Moreover, the variational principle enables us to dealwith the constraint by incorporating it in the set of admissible functions. In other words,we minimize a functional corresponding to (10) in the set$\mathcal{K}=\{u\in H_{0}^{1}(\Omega);\int_{\Omega}udx=V\}$ , (15)instead of minimizing in $H_{0}^{1}(\Omega)$ . In this way, we avoid the direct treatment of the nonlocalterm.One of the important points in the construction of approximate solution of the type(11) to free boundary constrained problems is to show the continuity of the correspondingminimizers and thus justify that the sets $\{u^{h}>0\}$ are open.44The discrete Morse flow method has been applied to the analysis of volume-constrainedevolutionary problems. A list of the main results follows.Theorem (parabolic problem, see [2])$u_{t}(t, x)-\triangle u(t, x)$ $=$ $f(t, x, u)+\lambda(t)$ , $\lambda=\frac{1}{V}\int_{\Omega}(u_{t}u+|\nabla u|^{2}-f(u)u)dx$ ,$u(t, x)$ $=$ $g(t, x)$ $t\in(0, T),$ $x\in\partial\Omega$ , (16)$u(0, x)$ $=$ $u_{0}(x)$ $x\in\Omega$ .If $u_{0}\in H^{1}(\Omega)\cap L^{\infty}(\Omega)$ then under certain conditions on $f$ and $g$ , there exists a func-tion $u\in H^{1}(Q_{T})$ , satisfying the boundary and initial conditions of (16) and the volumeconstraint (4) such that$\int_{0}^{T}\int_{\Omega}(u_{t}\phi+\nabla u\nabla\phi-f(u)\phi)dxdt=\int_{0}^{T}\int_{\Omega}\lambda\phi dxdt$ ,for each $\phi\in L^{2}(0, T;H_{0}^{1}(\Omega))$ . Furtherrnore, the solution is unique and Holder continuous.Theorem (hyperbolic problem, see [3])$u_{tt}(t,x)$ $=$ $\triangle u(t, x)+\lambda(u)$ , $\lambda=\frac{1}{V}\int_{\Omega}(u_{tt}u+|\nabla u|^{2})dx$ , (17)$u(t, x)$ $=g(t, x)$ $on$ $(0, T)\cross\partial\Omega$ , (18)$u(O, x)$ $=$ $u_{0}(x)$ , $u_{t}(0, x)=v_{0}(x)$ $in$ $\Omega$ . (19)There exists a weak solution whose properties depend on the regularity of initial data.For example, if $u_{0},$ $v_{0}$ belong to $H^{1}(\Omega)$ and $g=0$ , then weak solution is a function $u\in$$H^{1}(0, T;L^{2}(\Omega))\cap L^{\infty}(0, T;H_{0}^{1}(\Omega))$ satisfying $u(O)=u_{0}$ and the following identity for alltest functions $\phi\in C_{0}^{\infty}([0, T)\cross\Omega)$ with $\Phi=\int_{\Omega}\phi dx$$\int_{0}^{T}\int_{\Omega}(-u_{t}\phi_{t}+\nabla u\nabla\phi)dxdt-\int_{\Omega}v_{0}\phi(0)dx$$= \frac{1}{V}\int_{0}^{T}\int_{\Omega}(-u_{t}(u\Phi)_{t}+|\nabla u|^{2}\Phi)dxdt-\frac{1}{V}\int_{\Omega}u_{0}v_{0}\Phi(0)dx$ .In the end, let us consider a slow motion of a droplet attached to a plane with nonuni-form surface tension. This motion can be modeled by a parabolic free boundary equationwith nonlocal term:$u_{t}(t, x)=\Delta u(t, x)-\mu(x)\chi_{\epsilon}’(u(t, x))+\chi_{u>0}\lambda(t)$ in $Q_{T}$ . (20)The unknown function $u$ represents the shape of the drop and $\chi_{\epsilon}$ is a nondecreasingsmoothing of the characteristic function of the set $\{u>0\}$ . This term comes from therequirement of fixed contact angle at the free boundary of the drop. The coefficient $\mu$represents the changing surface tensions and $\lambda$ is a nonlocal term of the form$\lambda=\frac{1}{V}\int_{\Omega}(u_{t}u+|\nabla u|^{2}+\mu u\chi_{\epsilon}’(u))dx$,45originating in the volume constraint. For this type of problem we have the following result(see [4]).Theorem (parabolic free boundary problem) Let $u_{0}\in H_{0}^{1}(\Omega)\cap L^{\infty}(\Omega)$ and $\mu\in L^{\infty}(\Omega)$ .There exists a unique weak solution to the problem (20) that is Holder continuous in$[0, T]\cross\overline{\Omega}$ (if the initial datum is). The weak solution is defined only by test functions thathave compact support inside $\{u>0\}$ .Apart from theoretical results, the proposed method is very efficient in the numericalsolution of globally constrained evolutionary problems. In the numerical algorithm, time-discretized functional of the type (10) is discretized in space by FEM and minimized bythe steepest descent method in the constrained set (15). The volume constraint representsa linear condition which leads to simple projections on a hyperplane.We performed numerical computation for the problem (1)$-(3)$ with added gravityterm. This model expresses the motion of droplets on a slope. One can observe that thelargest droplet starts moving since its gravity overpowers the surface tension forces. Thenit catches up with the smaller droplets, fusing with them in turn.$\iota\cdot-$$t\cdot\cdot-$$m$$-$ $t\cdot\cdot-$ $-$References[1] K. Kikuchi, An approach to the construction of Morse flows for varriational function-als, in: J.-M. Coron, J.-M. Ghidaglia, F. H\’elein (Eds.), Nematics-Mathematicaland Physical Aspects, Nato Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Kluwer Acad.Publ., Dodrecht-Boston-London, 1991, pp. 195-198.46[2] K.\v{S}vadlenka-S.Omata, Construction of solution to heat type problems with volumeconstraint via the discrete Morse flow, Funkcialaj Ekvacioj 50, 2007, pp. 261-285.[3] K.\v{S}vadlenka - S.Omata, Mathematical modelling of surface vibmtion with volumeconstmint and its analysis, Nonlinear Analysis 69, 2008, pp. 3202-3212.[4] K.\v{S}vadlenka-S.Omata, Mathematical analysis of a constmined pambolic free bound-ary problem describing droplet motion on a surface, to appear in Indiana UniversityMathematics Journal.[5] A. Tachikawa, A variational approach to constructing weak solutions of semilinearhyperbolic systems, Adv. Math. Sci. Appl. 4, 1994, pp. 93-103.47

Discrete Morse flow for nonlocal problems (Problems in the Calculus of Variations and Related Topics)

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Discrete Morse flow for nonlocal problems (Problems in the Calculus of Variations and Related Topics)

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