Harterich, J.

Sakamoto, Kunimochi

Kyoto University Research Information Repository

Title Interface motions driven by reaction, diffusion and convection(Variational Problems and Related Topics)Author(s)Harterich, J.; Sakamoto, KunimochiCitation数理解析研究所講究録 (2008), 1591: 108-118Issue Date2008-04URL http://hdl.handle.net/2433/81612RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityInterface motions driven by reaction, diffusion andconvectionJ. H\"arterichInsititut f\"ur Mathematik I, Freie Uiversit\"at BerlinArnimallee 2-6, 14195 Berlin, GermanyK. SakamotoDepartment of Mathematical and Life Sciences, Hiroshima University1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan1. INTRODUCTION1.1. Interfaces driven by reaction-diffusion equations. Interfacial phenom-ena have been studied in terms of reaction-diffusion equations. In particularthe Allen-Cahn equation$(Arightarrow C)$ $\frac{\partial u}{\partial t}=\Delta u+g(u)$ , $t>0,$ $x\in \mathbb{R}^{N}$ ,where $g$ is a bi-stable reaction term, has been a widely used model to describevarious physical phenonena. Here, bi-stability of $g$ means that it is the minusof the derivative, $g(u)=-W’(u)$, of a double well potential $W(u)$ with non-degenerate wells located at $u=u_{\pm}$ . When $W(u_{-})\neq W(u_{+})$, a hyperbolicscaling gives rise to$\frac{\partial u}{\partial t}=\epsilon\Delta u+\frac{1}{\epsilon}g(u)$ , ($\epsilon>0$ is a scaling parameter)inwhich case the interface is drivenby the difference of potential values $W(u_{-})-$$W(u_{+})_{j}$$V=c$,where $V$ stands for the normal speed of the interface and $c$ is a constant deter-mined from $W$, i.e., $c\propto W(u_{+})-W(u_{-})$ .When the two wells have the same depth, then $c=0$ and hence the interfaceequation above does not give any information on the motion of the interface.In this case, we apply a parabolic scaling to (A-C) so that it has the $f_{0}u_{0}w\dot{m}g$formj$\frac{\partial u}{\partial t}=\Delta u+\frac{1}{\epsilon^{2}}g(u)$ .1591 2008 108-118 108The motion law of the interface for this equaton is the so called “mean curva-ture flow”, namely,$V=H$,where $H$ stands for the sum of principal curvatures of the interface. Such in-terface evolutions for reaction-diffusion equations have been established bymany authors (see [2]).The purpose of this paper is to investigate what happens to these interfaceequations when a convection term is added to (A-C).1.2. Reaction-diffusion with convection. We investigate interfacial phenom-ena for the following equation;(RDC) $u_{t}+divf(u)=\Delta u+g(u)$ , $(x,t)\in \mathbb{R}^{N}x(0,\infty)$ .This equation is derived as follows.When a physical quantity $u$ is carried by a flux $J$ with souroe term $g(u)$, thenthe balance equation is expressed as(BL) $\frac{\partial u}{\partial t}+divJ=g(u)$ .If the flux $J$ is represented by a (vector-valued) function $f$ : $\mathbb{R}arrow \mathbb{R}^{N}$ and thegradient of $u$,(Flux) $J=-\mu\nabla u+f(u)$ ($\mu>0$ viscosity),equation (BL) reduces to (RDC). In diffusive flow fields, the flux $f$ is supposedto originate from fluid flows, and therefore, should be coupled with Navier-Stokes equations goveming the flow field. We are considering here a simplifiedproblem without reference to such flow-field equations. From now on, we setthe viscosity equal to 1; $\mu=1$To describe the dynamics of (RDC) as $tarrow\infty$, we perfom a hyperbolicspatio-temporal scalin$g;(x,t)arrow(x/\epsilon,t/\epsilon)$ which reduces (RDC) to(1) $u_{t}+divf(u)=\epsilon\Delta u+\epsilon^{-1}g(u)$ ,where $e>0$ is a scaling parameter.Our objective below is to investigate the dynamics of (1) in the singular limit$\epsilonarrow 0$ .To consider the singular limuit $\epsilonarrow 0$ means that we are describ-ing the variation of $u$ over large spatial ranges as time $tarrow\infty$ inthe original system.109We work throughout under the following hypotheses.(H1): (i) $g\in C^{2}(\mathbb{R}),g(u_{*})=0$ at $u_{*}=u_{-},0,u+$’$g’(u_{-})<0$ , $g’(0)>0$ , $g’(u_{+})<0$ (bi-stable reaction tem).$(\ddot{u})f\in C^{3}(\mathbb{R},\mathbb{R}^{N})$Well-posedness of iitial value problem: Under the hypothesis(H1), the problem (1) with an imitial condition(2) $u(x,0)=\phi(x)\in BC_{unif}^{2}(\mathbb{R}^{N})$possesses a unique global (in time) solution living in $BC_{unif}^{2}(\mathbb{R}^{N})$ .This is proved by a standard way by using abstract theories forevolution equations (see [1], for example).2. PLANAR WAVESFirst thing to do is to study planar $ffavel\dot{i}g$ wave solutions. The travelingwave solution of (1) in ffie $\nu\in S^{N-1_{-}}direcUonju(x,t)=U($ $a_{\epsilon}^{e\cdot\nu-\cdot t})$ satisfies(3) $U”(z)+(s-f’(U(z))\cdot\nu)U’(z)+g(U(z))=0$ ,$z\in \mathbb{R}$ , $(’=d/dz)$ ,$U(\pm\infty)=u\pm$ , $U(0)=0$ ,where $s$ is the wave speed to be $detern\dot{u}ned$ togeher with the wave profile.The following result is obtained by a phase plane analysis.Proposition 2.1 ([3] and [5]). (i) For each direction $\nu\in S^{N-1}$ , there uniquelyexists a wave speed $s=s(\nu)$ for which the problem (3) has a unique heteroclinicorbit connecting $(u_{-},0)$ (at $z=-\infty$) and $(u_{+},0)$ (at $z=+\infty$).(ii) The wave speed $s(\nu)d\varphi mds$ on $\nu$ as smooth as the nonlinear terms $f’(u)$ and$g(u)$ do on $u$.(iii) The wave profile $Q(z;\nu)wi$th $Q(0;\nu)=0$ depends on $(z,\nu)$ as smooth asthe nonlinear terms $f’(u)$ and $g(u)$ do on $u$, and it is a (strictly) monotoneincreasingfunction of $z$ .(iv) If $f(u)$ is even and $g(u)$ is odd in $u$, then $Q(z)$ is an odd function of $z$ and thewave speed satisfies $s(\nu)\equiv 0$110An important feature is that the wave speed is orientation (direction) depen-dent. This anisotropy later gives rise to anisotropic mean curvature flows.It is interesting to note that the wave speed and the wave proMe are relatedas follows.$s( \nu)=\frac{(f(u_{+})-f(u_{-}))\cdot\nu}{u_{+}-u_{-}}+\frac{1}{u_{+}-u_{-}}\int_{-\infty}^{\infty}g(Q(z;\nu))dz$and$s( \nu)=\frac{G(u_{-})-G(u_{+})}{\int_{-\infty}^{\infty}Q_{z}(z;\nu)^{2}dz}+\frac{\int_{-\infty}^{\infty}Q_{z}(z;\nu)^{2}f’(Q(z;\nu))\cdot\nu dz}{\int_{-\infty}^{\infty}Q_{z}(z;\nu)^{2}dz}$ ,wheoe $G(u)$ is $\bm{t}$ anb-derivative of $g(u)$ . The Mt formula looks like agen-eralized $\ovalbox{\tt\small REJECT}$-Hugoniot condibon for viscous shocks for conservahon laws,while he second expoession oesembles the wave speed characterizabon for bi-stable reaction-diffusion equations, wiffi modfficabon in tems of he enbrewave proffie. It is also importrt to note ffiat $s(\nu)$ depends not only on heasymptotic states (whch is the case for reation-diffusion equations), but akoon the enMe viscous wave proffie.Lemma 2.2. In the nonlinear eigenvalue problem (3), if the nonlinearities are givm$by$$g(u)=-R(u-u_{-})u(u-u_{+})$ ,with $u_{-}<0<u_{+},$ $R>0$ ,$f(u)=\frac{1}{2}u^{2}a+ub$, a$,$ $b\in \mathbb{R}^{N}$ ,then, the wave speed $s(\nu)$ and wave profile $Q(z)$ are euplictly represented asfillows.$s( \nu)=\frac{u_{-}+u+}{4}\{a\cdot\nu-\sqrt{(a\nu)^{2}+8R}\}+b\cdot\nu$,$Q(z)= \frac{-u_{+}u_{-}+u_{+}u_{-}e^{-D(u-u-)z}+}{-u_{-}+u_{+}e^{-D(-u-)z}u+}$where $D$ is $d\phi ned$ by$D= \frac{\sqrt{(a\nu)^{2}+8R}-a\cdot\nu}{4}$ .The proof is the same as the case without convection term (see [6]). Onemayalso substitute these functions into (3) to directly verify the lemma.$\blacksquare$111Generically, we expect that the wave speed $s(\nu)$ vanishes on a subset in $S^{N-1}$which has codimension at least one. However, we have been unable to proveor disprove this expectation in general case. The set where $s(\nu)$ vanishes,$\mathcal{P}:=\{\nu\in S^{N-1}|s(\nu)=0\}$is called a set of pinned directions. When the wave speed is given as in Lemma2.2, then $\mathcal{P}$ is generically of codimension at least one. Althourgh this is the casefor the specific cases, we make the folowing hypothesis.(H2): The set of pinned directions $\mathcal{P}$ has codimension at least one in $S^{N-1}$3. RESULTSWe now give some of main results. There are two cases, one in which (H2)is valid, and the other where $\mathcal{P}=S^{N-1}$ .3.1. When codim $\mathcal{P}\leq 1$ . We call this case a hyperbolic scaling case.Theorem 3.1. Under the hypotheses $(Hl)$ and $(H2)$, we consider thefollowing Cauchyproblem(4) $\{\begin{array}{ll}u_{t}^{\epsilon} =\epsilon\Delta u^{e}-f’(u^{e})\cdot\nabla u^{e}+\epsilon^{-1}g(u^{\epsilon})u^{e}(x,0) =\phi^{e}(x).\end{array}$There exist twofunctions $\underline{u}_{0}^{e}(x)<\overline{u}_{0}^{\epsilon}(x)$ and a constant $T>0$ such that thefollowingstatement is true: If the initialfimction satisfies(x) $<\phi^{\epsilon}(x)<\overline{u}_{0}^{e}(x)$ ,then the solution $u^{e}(x,t)$ converges to a limit $u^{0}(x,t)= \lim_{\epsilonarrow 0}u^{\epsilon}(x,t)$ for almostall $(x,t)\in \mathbb{R}^{N}x[0,T]$ . The limit function $u^{0}(x,t)$ is a piece-wise constanfimction,assuming only two values u-and $u+\cdot$ The bulk regions$\Omega^{\pm}(t)$ $:=\{x\in \mathbb{R}^{N};u^{0}(x,t)=u\pm\}$are separated $w$ a hypersurface $\Gamma(t)$ , and the hypersurface (inteface) evolves accord-ing to the motion law$V=s(\nu)$ ,where $Vr\varphi resmts$ the normal velocity of the interface $\Gamma(t)$, and $\nu$ is a unit nomalvector on $\Gamma(t)$ pointing into the interior of the bulk region $\Omega^{+}(t)$ .Ifwe define $\epsilon$-dependent inteface $\Gamma^{e}(t)w$$\Gamma^{\epsilon}(t)=\{x\in \mathbb{R}^{N}|u^{e}(x,t)=0\}$ ,112then its motion law is governed by(5) $V^{\epsilon}=s(\nu^{\epsilon})$$+ \epsilon\{H^{\epsilon}(y,t)+\sum_{p,q=1}^{N}T_{pq}^{e}K_{\epsilon}^{pq}(y,t)\}$$+O(\epsilon^{2})$ ,where $f$ is the unit normal vector of $\Gamma^{\epsilon}(t)$ (pointing into the interior $of+reg\dot{w}n$),$s(\nu^{\epsilon})$ is the wave speed evaluated at $f,$ $H^{\epsilon}(y,t)$ is the sum of principal curvatures(mean curvature,for short) of $\Gamma^{\epsilon}(t)$ at $y\in\Gamma^{\epsilon}(t),$ $(T_{m}^{g})$ is a symmetric, positive semi-definite $N\cross N$ matrix depending only on $(f,g, \nu^{\epsilon}),$ $K_{\epsilon}^{m}$ is a symmetric tensor relatedto the secondfindamentalform of $\Gamma^{\epsilon}(t)$ .We note that $T>0$ in the statement above is determined by the time intervalwhere $V=s(\nu)$ has a smooth soluiton.Now, let us give explicit forms to the quantities appearing in the theorem.For this purpose, we use the ffavellin$g$ wave profile $Q=Q(z;\nu)$ . Let $P=$$P(z;\nu)$ be defined by$P(z)=Q_{z}(z; \nu)\exp(\int_{0}^{z}[s(\nu)-f’(Q(\tau;\nu)\cdot\nu]d_{\mathcal{T}})$ .We also let $\Gamma(t)$ be represented by $\gamma_{0}$ as follows.$\gamma_{0}$ : $\mathcal{M}\cross[0,T]\ni(y,t)\vdash;\gamma_{0}(y,t)\in\Gamma(t)$ ,where $\mathcal{M}$ is a reference manifold.Then, representing by $g=(g_{ij})$ and $h=(h_{ij})$ the first and second fundamentalforms of $\Gamma(t)$ , respectively, with $g^{-1}=(g^{ij})$ , we have$K^{pq}= \frac{\partial(\gamma_{0})^{p}}{\partial y^{j}}\frac{\partial(\gamma_{0})^{q}}{\partial y^{l}}\dot{?}^{k}h_{ks}g^{\iota l}$ ,$T=M_{0}^{-1}\int_{-\infty}^{\infty}\frac{1}{PQ_{z}}L(z)\otimes L(z)dz$ ,$L(z)= \int_{-\infty}^{z}P(z’)Q_{z}(z’)[\nabla_{\nu}s(\nu)-f’(Q(z’))]dz’$ ,$M_{0}= \int_{-\infty}^{\infty}P(z)Q_{z}(z)dz>0$ .It is clear from these formula that $T$ is positive semi-definite. Generically, weexpect that the matrix $T$ is positive $d\phi nite$, not only positive semi-definite. To113see this, let $a\in \mathbb{R}^{N}$ , then we have$a^{T}R=M_{0}^{-1}\int_{-\infty}^{\infty}\frac{1}{PQ_{z}}(L(z)\cdot a)^{2}dz\geq 0$.Therefore, $a^{T}R=0$ implies $L(z)\cdot a\equiv 0$, which in turn implies$(\nabla_{\nu}s(\nu)-f’(Q(z;\nu)))\cdot a\equiv 0$ .This is possible for non-zero a only when the vector $\nabla_{\nu}s(\nu)-f’(Q(z;\nu))$ isparallel to a constant vector for all $z\in \mathbb{R}$ . Generically, we do not expect thatthis should happen.On the other hand, when $f’(u)=b$ is a constant vector, then$s(\nu)=c+b\cdot\nu$where $c$ is the $travel\dot{i}g$ wave speed of$U_{zz}+cU_{z}+g(U)=0$ , $z\in \mathbb{R}$$\lim_{zarrow\pm\infty}U(z)=u\pm$ , $U(0)=0$ .Therefore, we have $\nabla_{\nu}s(\nu)-f’(Q(z;\nu))\equiv 0$, and hence $T=0$.3.2. When $s(\nu)\equiv 0$. We have been unable to give general conditions whichimply $s(\nu)\equiv 0$ on $S^{N-1}$ . In this subsection, therefore, we assume the folowingconditions are fulfilled.(H3): $f(u)$ is even and $g(u)$ is odd.Evidently, Proposition 2.1 says that (H3) implies $s(\nu)\equiv 0$ .Theorem 3.2. Under the hypotheses $tHl$) and $(H3)$, we consider thefillowing prob-lem;(6) $\{\begin{array}{ll}u_{t}^{\epsilon} =\Delta u^{e}-\epsilon^{rightarrow 1}f’(u^{\epsilon})\cdot\nabla u^{\epsilon}+\epsilon^{-2}g(u^{e})u^{e}(x,0) =\phi^{\epsilon}(x).\end{array}$$I7lere$ exist a class of initialfunctions $\phi^{e}$ and a constant $T>0$ such that the solution$u^{\epsilon}$ of (6) converges to a limitfor almost all $(x,t)\in \mathbb{R}^{N}\cross[0,T]$;$u^{0}(x,t):= \lim_{earrow 0}u^{\epsilon}(x,t)$ .The limifinction $u^{0}(x,t)$ is piecewise constant, taking on two values u-and $u+\cdot$The interface $\Gamma(t)$ separating two bulk regions $\Omega^{\pm}(t)$ $:=\{x\in \mathbb{R}^{N};u^{0}(x,t)=u\pm\}$evolves according to thefollowing motion law.(7) $V=H+ \sum_{p,q=1}^{N}T_{M}K^{pq}=\sum_{p,q=1}^{N}(\delta_{pq}+T_{pq})K^{n}$ ,114where$K^{pq}= \frac{\partial(\gamma_{0})^{p}\partial(\gamma_{0})^{q}}{\partial y^{j}\partial y^{l}}\dot{g}^{k}h_{ks}g^{\iota l}$ ,$T=M_{0}^{-1}\int_{-\infty}^{\infty}\frac{1}{PQ_{z}}L(z)\otimes L(z)dz$ ,$L(z)= \int_{-\infty}^{z}P(z)Q_{z}(z)f’(Q)dz’$,$A(z)=- \int_{0}^{z}f(Q(\tau;\nu))\cdot\nu d\tau$,$P(z)=e^{A(z)}Q_{z}(z)$ .As before, $T$ is positive semi-definite, and generically positive definite. It isalso of interest to note that $T$ introduces a kin$d$ of Riemanmianmetric (possiblydegenerate) in the ambient space $\mathbb{R}^{N}$ . If $T$ were the $N\cross N$ identity maUix, then$TK=tr(hg^{-1})$ would be the sum of principal curvatures of the interface.Proposition 3.1. $\mathfrak{R}$ sum $T_{pq}K^{n}$ is a weighted sum ofprincipal curvatures:$T_{m}K^{pq}=\sum_{i=1}^{N-1}w^{i}\kappa_{i}$ ,where $\kappa_{i}(i=1,2, \ldots, N-1)$ are principal curvatures of $\Gamma(t)$ and$w^{i}= \sum_{p,q=1}^{N}\sum_{j=1}^{N-1}T_{pq}\frac{\partial(\gamma_{0})^{p}}{\partial y^{1}}\frac{\partial(\gamma_{0})^{q}}{\partial y^{j}}g^{\dot{f}}$ .$I7_{l}er\phi ore,$ (7) is rewritten as(7) $V= \sum_{i=1}^{N-1}(1+w^{i})\kappa_{i}$ ,namely, the interface $\Gamma(t)$ is driven by an anisotropic man curvatureflow.By inspecbng the proof of Theorem 3.2, we obtain the existence result forequihibrium solutions of (6).Corollary 3.2. Let $\Gamma^{*}be$ anisotropicaly $\ovalbox{\tt\small REJECT}$ in the sense that$0= \sum_{i=1}^{N-1}(1+w^{i})\kappa_{i}$ on $\Gamma^{*}$ .115IfI” is non-degenerate in the sense that an elliptic linear operator $\mathcal{L}$ defined on $\Gamma^{*}does$not have O-eigenvalue, then there exis$ts$ a family of equilibirum solutions $u^{\epsilon}(x)$ of (6)for small $\epsilon>0$ so that$\lim_{arrow 0}u^{\epsilon}(x)=\{\begin{array}{ll}u+ x\in\Omega_{+}u_{-} x\in\Omega_{-}\end{array}$The linear operator $\mathcal{L}$ has the following explicit form.$\mathcal{L}A=\Delta^{\Gamma^{*}}A+T_{M}[\nabla_{\Gamma}\cdot(\nabla_{\Gamma^{*}}A)^{p}]^{q}$$+F\cdot\nabla_{\Gamma}\cdot A+.G\cdot\nabla_{\Gamma}^{(1)}A$$+[H^{(1)}-T_{n}(\nabla_{\Gamma}^{(1)}\nu^{p})^{q}]A$ ,where $\Delta^{\Gamma}$‘ is the Laplace-Beltrami operator on $\Gamma$“,$H^{(1)}= \sum_{j=1}^{N-1}(\kappa_{j})^{2}$ ,$\nabla_{\Gamma}\cdot A=\sum_{j,k=1}^{N-1}\frac{\partial\gamma^{*}}{\partial y^{j}}\dot{?}^{k_{\frac{\partial A}{\partial y^{k}’}}}$$\nabla_{\Gamma}^{(1)}A=\sum_{j,\iota,l,k=1}^{N-1}\frac{\partial\gamma^{*}}{\partial y^{j}}g^{j\epsilon}h_{\epsilon l}g^{lk}\frac{\partial A}{\partial y^{k}}$ ,and $F$ and $G$ are vector fields on $\Gamma$ to which we do not give explicit forms.However, we emphasize that $\mathcal{L}$ is the linearizahon of the right hand side of (7)around $\Gamma$“, relative to normal variations of hypersuface.4. SOME EXAMPLES4.1. Symmetnc nonlinearity. We deal with the fouowin$g$ specific nonlinear-$ity_{j}$$f(u)=\frac{1}{2}u^{2}a\in \mathbb{R}^{N}$ , $g(u)=-u(u^{2}-1)$Then we $obta\dot{i}j$ wave profile : $Q(z)=\tanh(Dz)$ and wave speed: $s(\nu)\equiv 0$ .Note that the v-dependency of the wave profile is only throught the quantity$D$ defined byTo describe the interface equation, we need to compute:116$Q_{q}$ $:= \frac{\partial Q}{\partial\nu^{q}}=-\frac{Da_{q}}{\sqrt{(a\nu)^{2}+8}}\frac{z}{\cosh^{2}(Dz)}$,$P=Q_{z}e^{A}=D(\cosh(Dz))^{-(+2)}\ddot{\tau}^{\nu}$$M_{0}= \int_{-\infty}^{\infty}e^{A(z)}Q_{z}^{2}(z)dz=D^{2}\int_{-\infty}^{\infty}(\cosh(Dz))^{-t_{\ddot{T}^{\nu}}+4)}dz$and$M_{0} T_{pq}=-\int_{-\infty}^{\infty}e^{A(z)}Q_{z}Q_{q}f_{p}’(Q)\cdot dz$$\frac{D^{2}*a_{q}}{\sqrt{(a\nu)^{2}+8}}\int_{-\infty}^{\infty}zt\bm{t}h(Dz)(\cosh(Dz))^{-t_{\ddot{T}^{\nu}}+4)}dz$$\frac{D^{2}a_{p}a_{q}}{\sqrt{(a\nu)^{2}+8}}(\frac{1}{a\cdot\nu+4D})\int_{-\infty}^{\infty}(coeh(Dz))^{-(\ddot{\tau}^{\nu}+4)}dz$.Therefore, by using $4D+a\cdot\nu=\sqrt{(a\nu)^{2}+8}$, we obtain$T_{pq}=\frac{a_{p}a_{q}}{(a\cdot\nu)^{2}+8}$ .On the other hand, a simple computation yields$T_{pq}K^{pq}=\frac{1}{2D^{2}+1}a\cdot\nabla_{\Gamma(t)}D$ ,and the interface equation is given by(8) $V=H(y,t)+ a\cdot\nabla_{\Gamma(t)}(\frac{Arc\tan(\sqrt{2}D)}{\sqrt{2}})$Here $D>0$ in $Q(z)=\tanh(Dz)$ means “steepness” of the wave profile, and(8) says that the tangential variation of the “steepness”Arctan $(\sqrt{2}D)$$\sqrt{2}$of the wave proMe is converted to the normal speed of the interface in thesingular limit. At a formal level, this kind of observation was first given in [4].117As before, the second term in the right hand side of (8) $T_{pq}K^{pq}$ is rewrittenand (8) reduces to(9) $V= \sum_{i=1}^{N-1}(1+w^{i})\kappa_{i}$ ,where$w^{i}= \frac{1}{(a\cdot\nu)^{2}+8}\sum_{j=1}^{N-1}(\frac{\partial\gamma_{0}}{\partial y^{j}}\cdot a)(\frac{\partial\gamma_{0}}{\partial y^{i}}\cdot a)\dot{?}^{i}$In other words, (9) is interpreted as a weighted mean curvature flow. Note that$w^{i}$ in (9) is $0$ when a is parallel to $\nu$ .Although the matrix $T$ originates from the first order differential operatordiv $f$, it exhibits a curvature effect (a second order differential operator), in thesingular limit.REFERENCES[1] D. Daners and P. Koch Medma, Abstract evolution equations, \mbox{\boldmath $\mu$}riodic $\psi oblms$ and $\varphi plica-$tion, Pibnan Research Notes 273 (1992).[2] $P$ de Mottoni and M. Schatzman, Geometrical evolution ofdeveloped interfaces, nans. AMS347 (1995), 1533-1589.[3] G. F. D. Duff, $Limit-\varphi cle\epsilon$ and rotated vectorfields, Ann. of Math. (2)57 (1953), 15-31.[4] H. Fan and S. Jin, Multi-dimensional viscous conservation laws with stiff souroe terms drivenby mean curvature and variation offront th\’ickness, Quarterly Appl. Math. LXI (4), (203),701-721.[5] J. Htirterich and C. Mascia, Front $Form\hslash\sigma n$ and Motion in Quasilinenr Parabolic Equations,preprint, 202.[6] J. D. Murray, Mathematical Biology, Sprmger (1989).118

Interface motions driven by reaction, diffusion and convection (Variational Problems and Related Topics)

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Interface motions driven by reaction, diffusion and convection (Variational Problems and Related Topics)

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