On a multiphase multicomponent model of biofilm growth

Transcription

1 Archive for Rational Mechanics and Analysis manuscript No. will be inserted by the editor Avner Friedman Bei Hu Chuan Xue On a multiphase multicomponent model of biofilm growth Abstract Biofilms are formed when free-floating bacteria attach to a surface and secrete polysaccharide to form an extracellular polymeric matrix EPS. A general model of biofilm growth needs to include the bacteria, the EPS, and the solvent within the biofilm region t, and the solvent in the surrounding region t. The interface between the two regions, Γ t, is a free boundary. In this paper, we consider a mathematical model, which consists of a Stokes equation for the EPS with bacteria attached to it, and a Stokes equation for the solvent in t and a different one for the solvent in t. The volume fraction of the EPS is another unknown satisfying a reaction-diffusion equation. The entire system is coupled nonlinearly within t, and across the free surface Γ t. We prove the existence and uniqueness of solution, with a smooth surface Γ t, for a small time interval. 1 Introduction Biofilms are defined as communities of microorganisms, typically bacteria, that are attached to a surface. The bacteria within the biofilms are usually embedded within a matrix consisting of polysacharides which is made by the bacteria and called extracellular polymeric substance EPS. The EPS shields the bacteria from attack, making them, in particular, resistant to therapeutic drugs [34. Biofilms account for over 80% of infections in the body. Examples include plaques on teeth and dental implants, infections of gastrointestinal tracks, urinary tracks, infections of eyes, nose, and ears, and infections of skin wounds [1. The bacterium Pseudomonas aeruginosa, an organism that can cause life threatening infections in diseases such as cystic fibrosis, is especially adept in resisting antimicrobial therapy [34. Improved understanding of the physical organization of a biofilm matrix may help in the development of new drugs aimed at disrupting the biofilm. Mathematical models of biofilm go back to the 1980 s. Two comprehensive reviews of the mathematical models, up to 2010, were published by Klapper and ockery [25, and Wang and Zhang [44. The most comprehensive models use two-phase flows to describe biofilm growth. The variables in these models include the volume fractions and velocities of the EPS and the solvent within the biofilm, and the velocity of the fluid surrounding the biofilm. These variables, as well as the concentration of bacteria and nutrients, satisfy a system of PEs within the biofilm, and a system of PEs outside the biofilm. These two systems are coupled through the common boundary Γ t, which is a free boundary. Klapper and ockery [25 explored the linear stability of the surface Γ t in case Γ t is initially flat and the linear perturbations are periodic. However, except for numerical results [6, there have been no rigorous mathematical results that address the questions of existence, uniqueness and properties of the full free boundary problem of the biofilm model. A. Friedman Mathematical Biosciences Institute, and epartment of Mathematics, Ohio State University, Columbus, Ohio afriedman@math.ohio-state.edu Bei Hu epartment of Applied and Computational Mathematics and Statistics, University of Notre ame, IN b1hu@nd.edu Chuan Xue Mathematical Biosciences Institute, and epartment of Mathematics, Ohio State University, Columbus, Ohio cxue@math.ohio-state.edu

2 In this paper, motivated by models of biofilms, we consider a general two-phase free boundary problem. In one phase, t, there is an incompressible viscous fluid, and in the other phase, t, there is a mixture of two incompressible fluids, which represent the viscous fluid and the polymeric network with bacteria attached to it associated with a biofilm; see Figure 1. The interface Γ = Γ t is a free boundary which evolves in time. Γ S Fluid G Biofilm Fig. 1 The Geometry of the model. In t, the fluid satisfies a Stokes system for the velocity and pressure of the fluid, and in t, the mixture satisfies a system of Stokes equations for the velocities of the fluid and the network and their common pressure. We prove that this system has a unique solution with smooth free boundary, for a small time interval 0 t T. There has been a lot of work in the area of Stokes systems. Regularity for solutions of elliptic and evolution problems are well known. L p estimates and Schauder estimates have been established for domains with smooth boundary as well as Lipschitz domains [29,30,23,39,40,42,41. When it comes to free boundary problems in the Stokes systems, issues of regularity need to be resolved in order to apply estimates for this type of systems. Free boundary problems similar to the one studied in this paper involving a Stokes system was studied in [8,7. That system of equations was derived from conservation of energy and, consequently, it was possible to formulate the free boundary condition in a weak form. This enabled the system to be solved in an appropriate Sobolev space with the free boundary conditions automatically built into the system. However, in general, free boundary problems for Stokes systems do not have a weak formulation. Instead, one may transform the free boundary to a fixed boundary either by a change of variables as in [21,33, or by introducing Lagrange variables, as in the work of Solonnikov [37,38. This latter approach was also used in [13 15 to prove local existence for a free boundary problem for a system which couples Stokes equation with several diffusion equations modeling tumor growth. Global existence for solutions with initial domain near a sphere was proved in [20,19,36 for viscous drops. The proofs in [20,19 use an expansion of the solution in terms of vector spherical harmonics. Symmetry-breaking bifurcations for a coupled system of Stokes equation and diffusion equations modeling tumor growth, were established in [16,17. A more complex system, modeling wound healing, which includes the Stokes equation coupled to diffusion and hyperbolic equations was considered in [18 where local existence was established. The present system for biofilm, however, is significantly more complex. It involves several Stokes flows in two domains with a free interface. Furthermore, on the free boundary, the conditions of continuity of velocities and forces are not standard. In addition, a coefficient θ n appears in the Stokes equations and in the boundary conditions at the free boundary, and θ n satisfies a diffusion equation coupled with the Stokes equations. The main result of this paper is a proof of existence and uniqueness of a smooth solution for a short time interval. The proof is based on the Schauder estimates and consists of three main steps presented in Sections 3-5. In Section 3, we consider the situation in which the fluids are in two fixed domains and θ n is a given function, and use the following procedure to prove the existence and uniqueness of solution for this fixed boundary problem: flatten the boundary and reflect the system in one domain locally into the other domain, use local Schauder estimates, and then a partition of unity to derive global Schauder estimates; apply the a priori estimates to a special system to establish existence; and finally by the method of continuity establish existence for the general system for the fixed boundary problem with given θ n. In Section 4, we use the approach of [18 to extend the results in Section 3 to the situation with moving boundaries but still with a given function θ n, using a fixed point theorem. In Section 5, we extend the analysis to include the variable θ n using a fixed point theorem once more. In Section 6 we show how our general result can be used to establish existence and uniqueness, in small time, for a general biofilm model. 2

3 2 The general mathematical model In this section, we state the general two-phase free boundary problem. We consider the geometry given in Figure 1, where a growing gel, modeled as a mixture of fluids, occupies a domain t outside a solid substrate G, and is surrounded by an incompressible fluid in a domain t. The interface Γ = Γ t between t and t is a free boundary which evolves in time. The PE system in t is given by, v 1 + v T 1 P 1 = f 1 in t, 2.1 v 1 = h 1 in t, 2.2 where v 1 and P 1 are the velocity and pressure of the fluid in the domain t, f 1 is the body force acting on the fluid, and h 1 is included for generality. In the t phase, we have a more complicated system of PEs, θ n t + θ nv 2 = ε θ n + G n in t, 2.3 [θ n v 2 + v T 2 θ n P 2 = f 2 in t, 2.4 [1 θ n η v 3 + v T 3 1 θ n P 2 = f 3 in t, 2.5 θ n v θ n v 3 = h 2 in t, 2.6 where θ n is the volume fraction of the polymer, 1 θ n is the volume fraction of the solvent, G n is the rate of mass conversion from solvent to polymer network, v 2 and v 3 are the velocities of the polymer and fluid, P 2 is the pressure of the mixture in the domain t, and f 2 and f 3 are the body force acting on the two components of the solution including the friction forces between the them, and h 2 is included for generality. We note that the total mass of the polymer network is not conserved if the conversion rate G n is nonzero. However, if h 1 = h 2 = 0, as we consider in the actual biofilm model in Sec. 6, the total mass of the solvent in t t and the polymer in t is conserved, reflecting conservation of mass in the entire domain. The boundary conditions are v 2 n + v T 2 n P 2 n v 1 n + v T 1 n P 1 n = g 1 on Γ t, 2.7 η v 3 n + η v3 T n P 2 n v 1 n + v1 T n P 1 n = g 2 on Γ t, 2.8 v 1 = θ n v θ n v 3 on Γ t, 2.9 v 1 = 0 on S and v 2 = v 3 = 0 on G, 2.10 θ n = 0 on t, 2.11 n where n is the outward normal, and g 1 and g 2 are additional forces acting on the moving interface Γ t, including, e.g., surface tension. The boundary conditions represents balance of forces and velocities across the free boundary. The velocity of the boundary Γ t in the normal direction is thus given by The initial condition for θ n is V Γ t = v n n on Γ t θ n t=0 = θ 0 in Existence and uniqueness of solution for the static reduced problem Given θ n, consider the static reduced problem for v 1, v 2, v 3, P : v 1 + v T 1 P 1 = f 1 in, 3.1 v 1 = h 1 in, 3.2 [θ n v 2 + v T 2 θ n P 2 = f 2 in, 3.3 [1 θ n η v 3 + v T 3 1 θ n P 2 = f 3 in, 3.4 θ n v θ n v 3 = h2 in, 3.5 3

4 with the boundary conditions v 2 n + v T 2 n P 2 n v 1 n + v T 1 n P 1 n = g 1 on Γ, 3.6 η v 3 n + η v T 3 n P 2 n v 1 n + v T 1 n P 1 n = g 2 on Γ, 3.7 v 1 = θ n v θ n v 3 on Γ, 3.8 v 1 = 0 on S and v 2 = v 3 = 0 on G. 3.9 In this section we prove that this problem has a unique classical solution, given that and η is a positive constant. f 1 C α, f 2 C α, f 3 C α, h 1 C 1+α, h 2 C 1+α, 3.10 g j C 1+α Γ, j = 1, 2, 3.11 S, G and Γ are in C 2+α, 3.12 θ n C 2+α, 0 < θ n < 1 in, 3.13 Remark. Note that θ n occurs in Equations After rewriting 3.5 in the form θ n v θ n v 3 = h 2 + h, where h includes terms with θ n, we see that θ n C 2+α is needed to ensure that the function h belongs to C 1+α. Structure of the proof. To prove this result, we consider the following parametrized problem: with the boundary conditions v 1 + λ v T 1 P 1 = f 1 in, 3.14 v 1 = h 1 in, 3.15 [θ n v 2 + λ v T 2 θ n P 2 = f 2 in, 3.16 [1 θ n η v 3 + λ v T 3 1 θ n P 2 = f 3 in, 3.17 θ n v θ n v 3 = h 2 in, 3.18 v 2 n + λ v T 2 n P 2 n v 1 n + λ v T 1 n P 1 n = g 1 on Γ, 3.19 η v 3 n + ηλ v T 3 n P 2 n v 1 n + λ v T 1 n P 1 n = g 2 on Γ, 3.20 v 1 = θ n v θ n v 3 on Γ, 3.21 v 1 = 0 on S and v 2 = v 3 = 0 on G, 3.22 with 0 λ 1. For easy reference we denote this system of equations and the boundary conditions, respectively, by L λ v = F, B λ v = G. We first prove uniqueness of solutions for L λ, B λ for all 0 λ 1 by deriving an energy equality. We then establish Schauder estimates as follows: i we consider the case of a planar interface, reflect one system across the interface and derive the Schauder estimates for the combined system in the reflected domain; ii using partition of unity {χ k } of a neighborhood of the interface and flattening the free boundary in a small region containing the support of χ k for each k, we use i to extend the Schauder estimates to general fixed domains. We next prove the existence of solutions for L 0, B 0. This is done by applying Lax-Milgram lemma to establish existence of a unique solution in H 1, and taking finite differences we extend the regularity to H 2, H 3, etc, and eventually to C 2+α. Finally, by a continuity method, we derive existence and uniqueness of solutions for the system which corresponds to the parameterized problem with λ = 1. 4

5 3.1 Uniqueness of solutions We shall prove that the only solution to the corresponding homogeneous problem of is zero. We multiply v 1,j to the j-th equation of 3.14 and apply integration by part. We obtain, Similarly, v 1 f 1 dx = v 2 f 2 dx = = = [ v 1,j [ v 1,j v 1,j i,j=1 [ v 2,j [ v 2,j θ n v2,j θ n i,j=1 v1,j + λ v 1,i P 1 dx x j x j v1,j + λ v 1,i n i P 1 n j ds x j v1,j + λ v 1,i x j dx + P 1 v 1 dx. v2,j θ n + λ v 2,i P 2 θ n dx x j x j v2,j + λ v 2,i n i P 2 n j ds x j v2,j + λ v 2,i dx + P 2 θ n v 2 dx. x j v 3 f 3 dx = = [ v 3,j [ v 3,j 1 θ n j=1 i,j=1 1 θ n η v3,j 1 θ n η v3,j Taking the sum of the above three equations, we obtain where I b is the sum of the boundary integrals + λ v 3,i x j 1 θ n P 2 x j v3,j η + λ v 3,i n i P 2 n j ds x j v3,j + λ v 3,i dx + P 2 1 θ n v 3 dx. x j I d = I b + I p I f, 3.23 dx I b = v1,j + λ v 1,i n i P 1 n j ds x j [ v 1,j [ v2,j + v 2,j θ n + λ v 2,i n i P 2 n j ds x j [ + v 3,j 1 θ n η j=1 v3,j + λ v 3,i n i P 2 n j ds, x j 5

6 and I b = Γ + I d = i,j=1 v 1,j v1,j + λ v 1,i dx + x j i,j=1 θ n v 2,j + 1 θ n η v 3,j v3,j + λ v 3,i dx, x j i,j=1 I p = P 1 v 1 dx + P 2 θ n v θ n v 3 dx, I f = v 1 f 1 dx + v 2 f 2 + v 3 f 3 dx. From we see that [ θ n v 2,j + 1 θ n v 3,j + = Γ Γ j=1 v1,j + λ v 1,i n i P 1 n j ds x j [ v2,j v 2,j θ n + λ v 2,i n i P 2 n j ds x j [ v 3,j 1 θ n Γ j=1 [ θ n v 2,j [ + θ n v 3,j Γ j=11 = θ n v 2 g θ n v 3 g 2 ds, Γ and from 3.2, 3.5, we obtain Notice next that i,j=1 = [ v1,j 1 + λ v3,j η + λ v 3,i n i P 2 n j ds x j v2,j + λ v 2,i n i P 2 n j x j 2 + λ v1,i v3,j η + λ v 3,i n i P 2 n j x j v1,j 2 + I p = v1,i i,j=1,i j x j P 1 h 1 + P 2 h 2 dx. 1 λ v1,i x j v2,j + λ v 2,i dx x j v1,j + λ v 1,i n i + P 1 n j x j 2 + i,j=1,i j and that similar formulas hold for v 2 and v 3. Hence we can rewrite I d in the form 2 I d = v1,i 2 v1,i 1 + λ + 1 λ + x j i,j=1,i j i,j=1,i j 2 + θ n v2,i 2 v2,i 1 + λ + 1 λ + x j i,j=1,i j i,j=1,i j θ n η v3,i 2 v3,i 1 + λ + 1 λ + x j i,j=1,i j v1,j η + λ v 1,i n i + P 1 n j x j λ v1,j + v 2 1,i, 2 x j λ v1,j + v 2 1,i dx 2 x j λ v2,j + v 2 2,i dx 2 x j λ v3,j + v 2 3,i dx. 2 x j i,j=1,i j 6

7 Given g i = h i = f i = 0, we then have I b = I p = I f = 0, so that I d = 0. If 0 λ < 1, then v k,i x j 0, a.e., i, j, k = 1, 2, 3. Since θ n 0, 1 in. Using the zero boundary condition we find that v k 0. If λ = 1, then from the vanishing of I d we only obtain v k,i 0, and v k,j + v k,i 0 a.e., i, j, k = 1, 2, x j By 3.22, on the fixed boundary S, we have v 1,i = 0, and in the domain, v 1,i = 0 a.e.. Therefore v 1,i 0 a.e. on the sets of points that can be connected by a line segment in x i direction to the boundary S. enote this = 0. Thus v 1,i = 0 a.e. in the portion of 1j 0 that is j i 1ij 0. coincides with. Otherwise, we continue this process and at each step increase the set set by 0 1i. Since v 1,j = 0 a.e. in 0 1j, v 1,i x j = v 1,j connected by a line segment in x j direction to 0 1i. enote this set by 0 1ij and let 1 1i = 1i If Γ is convex, then 1i 1 where v 1 = 0. Since Γ is continuously differentiable, after a finite number of steps we arrive at the set, so that v 1 = 0 a.e. in. Similarly, we can show that v 2 = v 3 = 0 a.e. in. Thus the homogeneous problem only has the zero solution for v. From the equation 3.1, we see that P = 0 up to a constant. 3.2 C 2+α a priori estimates For any vector valued functions v = v 1, v 2, v 3, we shall use the notation v H if all the components v j H, and we introduce the notation v H = v 1 2 H + v2 2 H + v3 2 H. For simplicity, we consider the case with λ = 1, but the estimates derived below hold true for 0 λ 1, although the details are a lit more complex. The local Schauder estimates in any domain bounded away from Γ is standard. In this section we shall derive estimates that are also valid in a neighborhood of Γ. It will be convenient to use the variables x, y, z instead of x 1, x 2, x Case I: Γ = {z = 0} and v 1 = v 2 = v 3 0 for x 2 + y 2 + z 2 d and small d We assume that Γ is given by {z = 0}, and θ n x, 1 θ n x are uniformly positive. To derive a priori estimates, we use the following reflection: Then v r 1 and P r 1 satisfy the following equations in, v r 1,ix, y, z = v 1,i x, y, z, i = 1, 2, v r 1,3x, y, z = v 1,3 x, y, z, P r 1 x, y, z = P 1 x, y, z, f r 1,ix, y, z = f 1,i x, y, z, i = 1, 2, f r 1,3x, y, z = f 3 x, y, z, h r 1x, y, z = h 1 x, y, z. v r 1 + v r 1 T P r 1 = f r 1, v r 1 = h r 1x, y, z. Using the reflected variables, the original system becomes v r 1 + v r 1 T P r 1 = f r 1 in, 3.25 v r 1 = h r 1x, y, z in, 3.26 [θ n v 2 + v T 2 θ n P 2 = f 2 in, 3.27 [1 θ n η v 3 + v T 3 1 θ n P 2 = f 3 in, 3.28 θ n v θ n v 3 = h 2 in,

8 with the boundary conditions v1,1 r x, y, 0 z + vr 1,3 x, y, 0 x + v 2,1x, y, 0 z + v 2,3x, y, 0 x + v 2,3x, y, 0 y = g 11 on Γ, 3.30 v1,2 r x, y, 0 + vr 1,3 x, y, 0 + v 2,2x, y, 0 z y z = g 12 on Γ, vr 1,3 x, y, v 2,3x, y, 0 + P 1 P 2 = g 13 z z on Γ, 3.32 v1,1 r x, y, 0 + vr 1,3 x, y, 0 + η v 3,1x, y, 0 + η v 3,3x, y, 0 = g 21 z x z x on Γ, 3.33 v1,2 r x, y, 0 + vr 1,3 x, y, 0 + η v 3,2x, y, 0 + η v 3,3x, y, 0 = g 22 z y z y on Γ, vr 1,3 x, y, 0 + 2η v 3,3x, y, 0 + P 1 P 2 = g 23 z z on Γ, 3.35 v1,1x, r y, 0 θ n v 2,1 x, y, 0 1 θ n v 3,1 x, y, 0 = 0 on Γ, 3.36 v1,2x, r y, 0 θ n v 2,2 x, y, 0 1 θ n v 3,2 x, y, 0 = 0 on Γ, 3.37 v1,3x, r y, 0 + θ n v 2,3 x, y, θ n v 3,3 x, y, 0 = 0 on Γ, 3.38 v 1 = v 2 = v 3 0 for x x z 2 d In the sequel, we drop the superscript r when there is no confusion. Ellipticity. We consider the ellipticity, in the sense of [2, p. 39, of the system for the variables v 1, P 1, v 2, v 3, P 2. The matrix l corresponding to the principal part of the operators in these equations is where and the determinant of l is L 1 Ξ T Ξ l Ξ = 0 0 θ n L 1 0 θ n Ξ T θ n L 1 1 θ n Ξ T, 0 0 θ n Ξ 1 θ n Ξ 0 Ξ = ξ 1, ξ 2, ξ 3, L 1 = Ξ 2 + Ξ T Ξ, LΞ = detl Ξ = 2η 3 θ 3 n1 θ n 3 Ξ 18 θ n + 1 θ n /η > 0, Ξ > 0. Therefore the system is elliptic. Supplementary condition. Since we consider a three dimensional problem, the supplementary condition is automatically satisfied [2, p. 39. Complementing boundary condition. In the following we show that the boundary conditions on Γ satisfy the Complementing Boundary Condition [2, p.42. We begin with some computations. Let m be any unit tangential vector on the boundary. Using the fact that m n = 0, we find that the roots τ + k of Lm + τn = 0 are τ + j = i, j = 1, 2,..., 9. As in [2, p.42, we set 6 M + τ = τ τ + j = τ i j=1 8

9 The matrix B Ξ corresponding to the principal part of the boundary operators is ξ 3 0 ξ 1 0 ξ 3 0 ξ ξ 3 ξ ξ 3 ξ ξ ξ ξ 3 0 ξ ηξ 3 0 ηξ 1 0 B Ξ = 0 ξ 3 ξ ηξ 3 ηξ ξ ηξ θ n θ n θ n θ n θ n θ n For n = 0, 0, 1, m = m 1, m 2, 0, we have m n = 0, Ξ = m + τn = m 1, m 2, τ. Using MATLAB symbolic computations we found that 8 B Ξadj l Ξ = τ j Qj, j=0 mod M + τ, with Q 0 a matrix with 9 rows and 11 columns. The determinant of the 9 9 submatrix Q 01 which consists the 1st, and 3rd-10th columns of Q 0, and the 9 9 matrix Q 02 which consists the 2nd-10th columns of Q 0, can be computed to be det Q 01 = m 2 η 27 θ 24 n 1 θ n θ n + 1 θ n /η 3 θ n + 1 θ n /η 9, det Q 02 = m 1 η 27 θ 24 n 1 θ n θ n + 1 θ n /η 3 θ n + 1 θ n /η 9. Since m m2 2 = 1, det Q 01 and det Q 02 cannot vanish simultaneously. Hence the rows of Q 0 are linearly independent. The system is said to satisfy the complimenting boundary condition if the rows of the matrix B Ξadj l Ξ are linearly independent mod M + τ; that is, if 6 C k k=1 j=0 5 τ j Qk j 0 mod M + τ, where Q k j is the k-th row vector of the matrix Q j, then all the constants C k are zero. Suppose not all the C k are zero. Then the row vectors of 5 j=0 τ j Qj are linearly dependent, for all τ, and, in particular, for τ = 0; that is, the row vectors of Q 0 are linearly dependent, which is a contradiction. Thus the complementing boundary condition on Γ has been verified.. Schauder estimates. Since the reflected system satisfies the supplementary condition and complementing boundary condition, we conclude by Theorem 9.3 on page 74 of [2, that the following Schauder estimates hold: v 1 C 2+α + C v j C 2+α + P 1 C 1+α + P 2 C 1+α j=2 f 1 C α + f i C α + h 1 C 1+α + h 2 C 1+α + i=2 j=2 g i C 1+α Γ +C v 1 + L v j L + P 1 L + P 2 L

10 3.2.2 Case II: Γ = {z = φx, y}, φ small in the C α norm In this section, we extend the results of Section to the case where Γ is given by z = φx, y with φ small in the C α norm, and the v i have bounded support. We flatten the boundary by taking x = x, y = y, and z = z φx, y. Under this change of variables, we have, for any function U x, y, z = Ux, y, z, U x = U x + U z φ x, U y = U y + U z φ y, U z = U z U xx = U xx + 2U xz φ x + U z z φ x 2 + U z φ xx, U yy = U yy + 2U yz φ y + U z z φ y 2 + U z φ yy, U xy = U xy + U xz φ y + U yz φ x + U z z φ x φ y + U z φ xy, U xz = U xz + U z z φ x, U zz = U zz. U yz = U yz + U z z φ y, We apply the formulas to the equations for the components of v 1, and for simplicity drop the primes. The equations for v 1 in the new variables are: 2v 11 xx + v 11 yy φ 2 x + φ 2 yv 11 zz 4φ x v 11 xz 2φ y v 11 yz 2φ xx + φ yy v 11 z + v 12 xy φ y v 12 xz φ x v 12 yz + φ x φ y v 12 zz φ xy v 12 z + v 13 xz φ x v 13 zz P 1 x + φ x P 1 z = f 11, v 11 xy φ y v 11 xz φ x v 11 yz + φ x φ y v 11 zz φ xy v 11 z + v 12 xx + 2v 12 yy +1 + φ 2 x + 2φ 2 yv 12 zz 2φ x v 12 xz 4φ y v 12 yz φ xx + 2φ yy v 12 z + v 13 yz φ y v 13 zz P 1 y + φ y P 1 z = f 12, or v 11 xz φ x v 11 zz + v 12 yz φ y v 12 zz + v 13 xx + v 13 yy φ 2 x + φ 2 yv 13 zz 2φ x v 13 xz 2φ y v 13 yz φ xx + φ yy v 13 z P 1 z = f 13, v 11 x φ x v 11 z + v 12 y φ y v 12 z + v 13 z = h 1, v 1 + v T 1 P 1 = f 1 + F v 1, P 1, φ + F 2 1 v 1, 2 φ in, v 1 = h 1 + H 1 φ, v 1 in, where F 1 1 C α C 2 v 1 Cα + P 1 Cα φ Cα, F 2 1 Cα C v 1 Cα 2 φ Cα, H 1 C 1+α C v 1 C 1+α φ C α + v 1 C α 2 φ C α. Similarly, under the new variables, [θ n v 2 + v T 2 θ n P 2 = f 2 + F v 2, P 2, φ + F 2 2 v 2, 2 φ, [1 θ n η v 3 + v T 3 1 θ n P 2 = f 3 + F v 3, P 2, φ + F 2 3 v 3, 2 φ. θ n v θ n v 3 = h 2 + H 2 v 2, v 3, φ, The boundary conditions are transformed in a similar way. 10

11 We now use the reflections as in Section 3.2.1, and thus derive that the following estimates for any solution of the system: v j C 2+α + j=1 P j C 1+α j=1 { C f i C α + φ C α 2 v j C α + 2 φ C α v j C α + h i C 1+α + g i C 1+α + φ C α v j C 1+α + v 1 L + } v j L + P 1 L + P 2 L. j=2 Using the assumption that φ C α is small, and the interpolation inequality v j C 1+α ε v j C 2+α + C ε v j L, and going back to the original domain, we obtain the Schauder estimates We note that the constant C depends on θ n Case III: arbitrary Γ = {z = φx, y} Let χ k k = 1,..., N be a partition of unity of, such that χ 1,..., χ N1 form a covering of Γ, χ N1+1,..., χ N2 form a covering of G, and the support of the remaining χ k lie in the interior of. The support of each χ k is taken as a ball B k with a small radius < d. By translation and rotation, we may assume that B k Γ, for any 1 k N 1, has the form z = ψx, y where ψ C α is small, as in Section Then the Schauder estimates can be applied in this ball. Similarly one can derive Schauder estimates in the balls which cover G. The Schauder estimates hold also in each ball B k for k > N 2, and they also hold for any compact subset 1 of which does not intersect Γ. efine u k = χ k u, v k = χ k v, w k = χ k w, P k j v j = = χ kp j, so that N vj k, j = 1, 2, 3, and Pj k = k=1 For each k, we have the following equations, N Pj k, j = 1, 2. k=1 v k 1 + v k 1 T P k 1 = χ k f χ k v 1 + χ k v 1 + χ k h 1 + P 1 + χ k 2 v 1 + v T 1 in, v k 1 = χ k h 1 + χ k v 1 in, [θ n v k 2 + v k 2 T θ n P k 2 = χ k f 2 + θ n 2 χ k v 2 + θ n χ k v 2 +θ n χ k v 2 + v T 2 + χ k θ n v 2 + χ k θ n v 2 + θ n P 2 χ k in, [1 θ n η v 3 + v T 3 1 θ n P 2 = χ k f 3 + η1 θ n 2 χ k v 3 + η1 θ n χ k v 3 +η1 θ n χ k v 3 + v T 3 + η χ k 1 θ n v 3 +η χ k 1 θ n v 3 + η1 θ n P 2 χ k in, θ n v θ n v 3 = χ k h 2 + χ k θ n v θ n v 3 in, with corresponding boundary conditions whose right hand side may involve χ k. 11

12 By applying 3.42 we obtain, for each k, vj k C 2+α + j=1 Pj k C 1+α j=1 C f i C α + i h i C 1+α + g i C 1+α +C χ k L v j C 1+α + P j C α + 2 χ k L v j C α Summing over k, and combining the Schauder estimates for v i, P i, we obtain v 1 C 2+α + v j C 2+α + P 1 C 1+α + P 2 C 1+α j=2 C f 1 + Cα f i Cα + h 1 C 1+α + h 2 C 1+α + i=2 j=2 g i C 1+α Γ +C v 1 C 1+α + v j C 1+α + P 1 Cα + P 2 Cα Using again interpolation, we again obtain the estimate Noting that the solution is unique, up to constants for P j, we proceed to show that the terms involving the L norms on the right-hand side of 3.42 can be dropped when the 1 + α norm of P j is replaced by α norm of P j. Clearly, the estimate 3.42 is valid when P j is replaced by P j µ, for any constant µ. We take a point M Γ and set µ = P 1 M. Then, by Poincaré s inequality, and, by 3.19, P 1 µ L C P 1 L, P 2 µ L P 2 P 2 M L + P 2 M P 1 M L From 3.42 it then follows that C P 2 L + C g 1 L + v 1 L + v 2 L v 1 C 2+α + v j C 2+α + P 1 Cα + P 2 Cα j=2 C f 1 C α + f i C α + h 1 C 1+α + h 2 C 1+α + i=2 +C v 1 L + v j L + P 1 L + P 2 L. j=2 g i C 1+α Γ 3.46 Next, we shall prove, by contradiction, that the L norms of the v i can be eliminated by the right-hand side of 3.46, so that v 1 C 2+α + v j C 2+α + P 1 Cα + P 2 Cα j=2 C f 1 C α + f i C α + h 1 C 1+α + h 2 C 1+α + i=2 g i C 1+α Γ Assume that this is not true. Then there exist sequences of data fi n, hn i, gn i bounded in their respective norms, and solutions vj n such that Q n {left-hand side of 3.47 corresponding to vj n}. Hence Q0 n v1 n L + 12

13 3 j=2 vn j L + P 1 n L + P 2 n L. We normalize the solution and data by Q 0 n. Since the system of equations is linear, we get fi n, hn i, gn i 0, so that, by 3.47, a subsequence of the solutions v1 n, vn 2, vn 3, P 1 n, P 2 n converges uniformly in their respective domains to a solution v 1, v 2, v 3, P 1, P 2 which is, by uniqueness, identically zero. Since, however v 1 L + 3 j=2 v j L + P 1 L + P 2 L = 1, this is a contradiction. We note that the above proof is valid for any 0 λ 1 and the constant C in 3.47 can be chosen to be independent of λ. 3.3 C 3+α a priori estimates The arguments used to derive the C 2+α Schauder estimate can also be applied to show that if Γ C 3+α, θ n C 3+α, f j C 1+α, g i, h i C 2+α in their respective sets, then the following C 3+α Schauder estimate holds: v 1 C 3+α + v j C 3+α + P 1 C 1+α + P 2 C 1+α j=2 C f 1 C 1+α + f i C 1+α + h 1 C 2+α + h 2 C 2+α + i=2 g i C 2+α Γ Existence of solution for λ = 0 In this section, we take λ = 0, and prove that if Γ is in C, f 1, h 1 C, f 2, f 3, h 2 C, and g 1, g 2 C Γ, then there exists a weak solution, v 1, v 2, v 3, P 1, P 2 H 1 H 1 H 1 L 2 L 2 to the system with boundary conditions Next we shall improve the regularity and show that the solution is in C 2+α. Finally, we shall prove the existence of solution in C 2+α under the assumptions Existence of weak solution We first consider the case that h 1 = h 2 = 0 in, and g 1 = g 2 = 0 on Γ. The general case will be considered later. We introduce the linear space H of vectors φ 1, φ 2, φ 3 H where each φ i is a 3 vector, as follows, H = {φ 1 H 1, φ 2 H 1, φ 3 H 1 : φ 1, φ 2, φ 3 satisfies 3.15, 3.18, 3.21, 3.22}. We define a norm H on H by and an inner product by φ 1, φ 2, φ 3 H = φ 1 2 H 1 + φ 2 2 H 1 + φ 3 2 H 1 1/2 < φ 1, φ 2, φ 3, ψ 1, ψ 2, ψ 3 >=< φ 1, ψ 1 > H 1 + < φ 2, ψ 2 > H 1 + < φ 3, ψ 3 > H 1. One can easily show that H is a Hilbert space. We multiply the j-th equation of 3.14, 3.16 and 3.17 by vector functions φ 1,j, φ 2,j, and φ 3,j, then take the sum over j, and integrate over the domain for 3.14, and over the domain for 3.16 and After integration by parts, we obtain, [ v1,j φ 1 f 1 dx = φ 1,j P 1 dx x j v1,j = n i P 1 n j ds [ φ 1,j φ 1,j i,j=1 13 v1,j dx + P 1 φ 1 dx.

14 φ 2 f 2 dx = = [ v2,j P 2 φ 2,j θ n θ n dx x j v2,j n i P 2 n j ds [ φ 2,j θ n φ2,j θ n i,j=1 v2,j dx + P 2 θ n φ 2 dx. φ 3 f 3 dx = = [ φ 3,j [ φ 3,j 1 θ n j=1 i,j=1 1 θ n η v3,j 1 θ n η φ3,j 1 θ n P 2 x j v3,j η n i P 2 n j ds v3,j dx dx + P 2 1 θ n φ 3 dx. We take the sum of the above three equations, and obtain the relation I d = I b + I p I f, 3.49 where I b is the sum of the boundary integrals and If φ 1, φ 2, φ 3 H, then [ v1,j I b = φ 1,j n i P 1 n j ds [ v2,j + φ 2,j θ n n i P 2 n j ds [ v3,j + φ 3,j 1 θ n η n i P 2 n j ds, j=1 φ 1,j v1,j φ 2,j I d = dx + θ n i,j=1 i,j=1 + 1 θ n η φ 3,j v3,j dx, i,j=1 I p = P 1 φ 1 dx + P 2 θ n φ 2 + θ 3 φ 3 dx, I f = φ 1 f 1 dx + φ 2 f 2 + φ 3 f 3 dx. I p = 0, v2,j dx 14

15 and I b = Γ + = [ v1,j θ n φ 2,j + 1 θ n φ 3,j n i P 1 n j ds j=1 Γ j=1 Γ j=1 + = Γ [ v2,j φ 2,j θ n n i P 2 n j ds + Γ θ n φ 2,j [ v2,j n i P 2 n j 1 θ n φ 3,j [η v1,j v3,j Γ j=1 θ n φ 2 g θ n φ 3 g 2 ds = 0. efine the bilinear form B[φ, v = I d = i,j=1 + By the Cauchy-Schwartz inequality, Also, B[v, v = C i,j=1 v1,i i,j=1 φ 1,j j=1 n i + P 1 n j ds v1,j n i P 2 n j v1,j 1 θ n η φ 3,j 2 dx + dx + v3,j [ v3,j φ 3,j 1 θ n η n i P 2 n j ds i,j=1 + P 1 n j ds θ n φ 2,j v2,j dx dx B[φ, v C φ H v H θ n i,j=1 v2,i x j x j v 1 2 L 2 + v 2 2 L 2 + v 3 2 L 2 2 dx + 1 θ n η, C > 0. i,j=1 v3,i x j 2 dx We can then use the Poincaré s inequality since v = 0 on the fixed boundary, by 3.22, to deduce that B[v, v C v 2 H, 3.52 with another positive constant C. In view of 3.51 and 3.52, we can apply the Lax-Milgram lemma, to conclude that there exists a unique element v H such that B[φ, v = I f φ, φ H Next we show that there exist scalar functions P 1 L 2, P 2 L 2 such that where v is the weak solution of 3.53, and B[φ, v = I p φ I f φ, φ Ĥ, 3.54 Ĥ = {φ 1 H 1, φ 2 H 1, φ 3 H 1 : φ 1, φ 2, φ 3 satisfies 3.21, 3.22}. This will prove that v 1, v 2, v 3, P 1, P 2 is a weak solution in case the h i and g i are identically zero. We will need Propositions 1.1 and 1.2 ii of [43, which we restate as a lemma: Lemma 1 i Let be an open set of R n and f = f 1,..., f n, f i, i = 1,..., n. A necessary and sufficient condition that f = p, for some p in, is that f, ν = 0, for all ν C0 with ν = 0. ii Let be a bounded Lipschitz open set in R n. If a distribution p has all its first derivatives i p, 1 i n, in H 1, then p L 2 and p L 2 /R c p H. 1 15

16 In the statement i, the notation f, ν denotes the application of the distribution f to a test function ν. To any Φ H0 1 Γ, we correspond a vector φ by φ 1 = Φ, φ 2 = φ 3 = Φ, φ = φ 1, φ 2, φ 3. It is clear that Φ = 0 if and only if φ 1 = 0 and θ n φ 2 +1 θ n φ 3 = θ n +1 θ n Φ = Φ = 0. We define the distribution F by FΦ = B[φ, v + I f φ, where φ = φ 1, φ 2, φ 3. Then FΦ = I p Φ = 0 if Φ C0 P such that Further, since Γ with Φ = 0. Thus, by Lemma 1 i, there exists a distribution F = P. FΦ B[φ, v + I f φ C Φ H 1 Γ, where C is a positive constant depending on f i, v i, θ n, we deduce that F = P H 1 Γ, and by Lemma 1 ii, we conclude that P L 2 Γ. We next define P 1 = P, P 2 = P, so that P 1 L 2, P 2 L 2, and proceed to show that P 1, P 2 is a solution to 3.54, i.e., for any φ Ĥ, φ i C, the equation 3.54 holds. We introduce a function Φ H0 1 Γ by Φ = φ 1, Φ = θ n φ θ n φ 3. The conditions 3.21, 3.22 imply that Φ H0 1 Γ. We define φ = Φ, Φ, Φ ; then ˆφ = φ φ H again using θ n Φ + 1 θ n Φ = Φ. Since Φ H0 1 Γ, we obtain B[ φ, v + I f φ = FΦ =< P, Φ >= and since ˆφ H, we also have, by 3.53, Therefore, for φ = ˆφ + φ, we have B[ ˆφ, v + I f ˆφ = 0. P Φ dx = I p φ, B[φ, v + I f φ = B[ ˆφ, v + I f ˆφ + B[ φ, v + I f φ = 0 + I p φ = I p φ, and so the assertion 3.54 holds. To summarize, we proved that there exist a weak solution v 1, v 2, v 3, P 1, P 2 H 1 H 1 H 1 L 2 L 2 to the static reduced problem, given h 1 = h 2 = 0 in, g 1 = g 2 = 0 on Γ. We next consider the case g 1 = 0, g 2 = 0 on Γ, but h 1, h 2 are arbitrary. We can construct functions w 1 H 1, w 2 H 1 such that Indeed, if we let w 1 = h 1 in, w 2 = h 2 in, 3.55 w 1 = w 2 on Γ, w 1 = 0, w 2 = 0 on the fixed boundaries. w = w 1 in, w = w 2 in, h = h 1 in, h = h 2 in, then the problem of constructing the w i is equivalent to solving w = h in Γ, w = 0 on S G. 16

17 Given h L 2 Γ, one can solve the Stokes equation to obtain w H0 1 Γ ; see [31. By subtracting w 1, w 2, w 2 from v 1, v 2, v 3, we can eliminate h 1, h 2, with new functions f i and g i. Finally, we consider the case where also g 1, g 2 are arbitrary. We want to introduce functions u 1, u 3, Q 1, Q 2, and subtract them from v 1, v 3, P 1, P 2, in a way that g 1, g 2 can be eliminated without changing any equations except for the functions f i. We take u 1, u 3, Q 1, Q 2 such that u 1 = 0 in, 1 θ n u 3 = 0 in, u 1 n + Q 1 n = g 1 on Γ, 3.56 η u 3 n Q 2 n = g 2 g 1 on Γ, u 1 = 1 θ n u 3 on Γ, u 1 = 0 on S u 3 = 0 on G. The above system is under-determined, and we construct a solution in two steps as follows. We begin by constructing a curvilinear orthogonal coordinate system q 1, q 2, q 3 with scalar factors h 1, h 2, h 3 in an ε-neighborhood of Γ, where ε is taken small such that the coordinate system is well-defined, with Γ = {q 3 = 0}, and we construct the solution in the region q 3 ε. Writing out the above equations except the last two in this new coordinate system near Γ, we have [ 1 u11 h 2 h 3 u 1 = h 1 h 2 h 3 q 1 + u 12h 1 h 3 q 2 + u 13h 1 h 2 q 3 = 0, θ n u 3 [ 1 1 θn u 31 h 2 h 3 = h 1 h 2 h 3 q θ nu 32 h 1 h 3 q θ nu 33 h 1 h 2 q 3 = 0, u 11 h 3 q 3 u 13 h 3 h 1 h 3 q 1 = g 11 on Γ, u 12 h 3 q 3 u 13 h 3 h 2 h 3 q 2 = g 12 on Γ, u 13 h 3 q 3 + u 11 h 3 h 1 h 3 q 1 + u 12 h 3 h 2 h 3 q 2 Q 1 = g 13 on Γ, 3.61 η 1 h 3 u 31 q 3 η u 33 h 1 h 3 h 3 q 1 = g 21 g 11 on Γ, 3.62 η 1 h 3 u 32 q 3 η u 33 h 2 h 3 h 3 q 2 = g 22 g 12 on Γ, 3.63 η 1 u 33 h 3 q 3 + η u 31 h 3 h 1 h 3 q 1 + η u 32 h 3 h 2 h 3 q 2 Q 2 = g 23 g 13 on Γ, 3.64 u 11 = 1 θ n u 31, u 12 = 1 θ n u 32, u 13 = 1 θ n u 33 on Γ If we take u 1 = u 3 = 0 on Γ, then 3.65 is automatically satisfied, and the boundary conditions can be simplified to 1 u 11 h 3 q 3 = g 11 on Γ, u 12 h 3 q 3 = g 12 on Γ, u 13 h 3 q 3 Q 1 = g 13 on Γ, 3.68 η 1 h 3 u 31 q 3 = g 11 on Γ, 3.69 η 1 h 3 u 32 q 3 = g 12 on Γ, 3.70 η 1 h 3 u 33 q 3 Q 2 = g 13 on Γ,

18 In the region { q 3 ε}, we define q 3 u 11 = g 11 q 1, q 2 ζq 3 u 12 = g 12 q 1, q 2 ζq 3 0 q 3 0 h 3 q 1, q 2, τdτ, h 3 q 1, q 2, τdτ, where ζq 3 is a cutoff function that satisfies ζq 3 = 1 if q 3 ε/4, and ζq 3 = 0 if q 3 ε/2. We then solve 3.57 for u 13, u 13 = 1 q 3 u11 h 2 h 3 h 1 h 2 0 q 1 + u 12h 1 h 3 q 2 dτ, and, finally, we use 3.68 to define Q 1. We also define u 3, Q 2 in { q 3 ε } in a similar way. We next extend the solution u 1, and using a similar method, we can extend u 3. We take a domain with smooth boundary, and \{q 3 < ε} \{q 3 < 3ε/4}; then u 1 is already defined on \Γ. We redefine and extend u 1 into \ by solving the following problem, u 1 = 0 in \, u 1 = u 1 \Γ on \Γ, 3.72 u 1 = 0 on S. The problem 3.72 can be solved in a similar way as before. The functions Q 1 and Q 2 can be extended in any way, provided that we preserve their regularity. The functions u 1, u 3, Q 1, Q 3 defined above thus form a solution to the problem Hence the case of arbitrary g 1, g 2 can be reduced to the case of g 1 = g 2 = 0. In conclusion, we have proved that there exists a weak solution v 1, v 2, v 3, P 1, P 2 to the static reduced problem with λ = 0, and v 1 H 1, v 2, v 3 H 1, P 1 L 2, P 2 L Regularity of solution In this section we prove that the solution v 1, v 2, v 3, P 1, P 2 is in C if the boundaries, S, G and the inhomogeneous data are in C. Interior regularity. Consider the bilinear form B[φ, v = I f φ, φ H. Given an open set V with V, we take open sets W, Ŵ such that V W, W Ŵ, Ŵ, and select a smooth cutoff function ζ such that ζ 1 in V, ζ 0 in \W, 0 ζ 1. We take φ 1 = ζ 2 v 1 h h χ h h, φ 2 = 0, φ 3 = 0, where 0 < h < min{d, W, d W, Ŵ }, and a function χ h which satisfies u h = ux + he j ux, h χ h = ζ 2 v 1 h in, χ h = 0 in \Ŵ. The function χ h can be chosen such that χ h H 1 C v h 1 L 2, where C does not depend on h. 18

19 If we substitute φ into the bilinear form 3.50, we obtain B[φ, v = = = i,j=1 i,j=1 i,j=1 [ i ζ 2 v 1,j h h χ h h j i v 1,j dx i [ ζ 2 v 1,j h + χ h j i v 1,j h dx 2ζ i ζ v 1,j h i v 1,j h + ζ 2 i v 1,j h 2 + i χ h j i v 1,j h dx. Also, I f φ = = j=1 j=1 [ ζ 2 v 1,j h h χ h h j f1,j dx [ ζ 2 v 1,j h + χ h j f1,j h dx. Since we have Thus v 1 h L 2 C v 1 H 1, χ h L 2 C v 1 H 1, f h 1 L2 Ŵ C f 1 H 1, B[φ, v 1 2 v 1 h 2 L 2 V C 2 v 1 2 H 1 I f C v 1 2 H 1 + f 1 2 H 1. v h 1 L 2 V C v 1 H 1 + f 1 H 1, and, by Lemma 15.3 of [12, we conclude that v 1 H 1 V, i.e., v 1 Hloc 2. Similarly, we can show that v 2, v 3 Hloc 2. For any φ with φ 1 H0 1V, φ 2 = φ 3 = 0, P1 h φ 1 dx = B[φ, v h + f1 h φ 1 dx C v 1 H 2 + f 1 H 1 φ1 H 1. Thus by [43, we have P1 h L 2 V C, and by Lemma 15.3 of [12, P 1 Hloc 1. Similarly, we can show that P 2 Hloc 1. Boundary regularity. We next show that the velocities v i are in H 2 along Γ. A simpler argument can establish the regularity near the fixed boundaries, and it is omitted here. We use partition of unity to cover the boundary Γ by a finite number of balls B ε with radius ε. We introduce new variables x = x, y = y, z = z ζx, y to flatten the boundary about each of the local coordinate systems. The radius ε is taken small enough so that the variational formulation of the transformed system is coercive. Then we can use the same method as for the interior estimates to show that second order derivatives of x v i, y v i, and the first order derivatives x P i, y P i are in L 2 in their respective domains. erivatives in the other directions are obtained from the transformed equations. In particular, from the equation for v 1, P 1, we obtain, in B ε, 1 + 2ζ 2 x + ζ 2 yv 11 zz + ζ x P 1 z =: Y 1 L 2 B ε, 1 + ζ 2 x + 2ζ 2 yv 12 zz + ζ y P 1 z =: Y 2 L 2 B ε, 2 + ζ 2 x + ζ 2 yv 13 zz P 1 z =: Y 3 L 2 B ε, ζ x v 11 zz ζ y v 12 zz + v 13 zz =: Y 4 L 2 B ε. 19

20 Notice that this is a set of four linear equations for v 11 zz, v 12 zz, v 13 zz, P 1 z, and the coefficients matrix is non-singular. Therefore we can solve the distributions v 11 zz, v 12 zz, v 13 zz, P 1 z as linear combinations of Y i, i = 1, 2, 3, 4, and thus conclude that v 11 zz, v 12 zz, v 13 zz, P 1 z are in L 2 B ε. The argument for 2 z z v 2, 2 z z v 3, z P 2 is similar. By differentiating the equation for v i and applying the above argument we can show that v i are in H 3 loc. The H 3 regularity near the boundary is established, as above, by differentiating first in the tangential directions and then in the normal directions. Similarly, P 1 is in H 2. Repeating this process step-by-step, we conclude that all the derivatives of the v i are in H 2 up to the boundary, provided the f i, h i, g i, θ n, and the boundary are all in C. Hence, v i and P i are in C if the data are in C Classical solution So far we have proved existence and uniqueness in the case λ = 0. Using the Schauder estimates for smooth data we can establish, by the standard method of continuity in the parameter λ, the existence and uniqueness of a solution for each λ, 0 < λ 1. Finally, by approximating Γ and the inhomogeneous data in the appropriate Hólder norms by data for which the right-hand side of 3.47 is uniformly bounded, we derive existence and uniqueness of a solution for λ = 1 with Γ in C 2+α and data for which the right-hand side of 3.47 is finite; the solution satisfies the estimate Existence and uniqueness of solution for the reduced free boundary problem In this section we prove existence and uniqueness of solutions for the reduced free boundary problem with Γ as a moving boundary at the normal velocity V Γ, but with θ n as a given function. The free boundary Γ = Γ t is moving by the velocity v n of the network, so that If we represent Γ t as Ψt, x, y, z = 0, then the equation for Γ t is, V Γ = v n n. 4.1 Ψ t + v n Ψ = 0 on Γ t, n = Ψ Ψ on Γ t. Structure of the proof. We use a method similar to our paper [18. We introduce a parameterization of a free boundary x = Xλ, t and denote by hλ, t the distance from Xλ, 0 to Xλ, t. For any family Xλ, t we solve the fixed boundary problem and use the free boundary condition to define a corresponding function hλ, t. We shall prove that the mapping hλ, t hλ, t is a contraction mapping. To do this we use the hyperbolic equation that hλ, t satisfies to derive a priori estimates. In the following we choose orthogonal curvilinear coordinates λ = λ 1, λ 2 on the initial free boundary X 0, where λ varies in a region Λ, and denote by e n λ, e 1 λ, e 2 λ the orthonormal system along the initial boundary X 0 λ, with e n λ the outward normal, and with e 1 λ, e 2 λ in the tangential plane. We assume that the free boundary can be represented in the form x = Xt, λ, λ Λ. Set 0 = 0, Γ 0 = Γ 0, so that Γ 0 is given by x = X0, λ = X 0 λ. We assume that X 0 λ is in C 3+α Λ, 4.2 for some α 0, 1; if we use spherical coordinates λ = θ, φ, then we need to require that X 0 λ C > 0, λ Λ. We also assume that for 0 t T, θ n, t C 2+α Λ, f j, t Cα Λ, g j, t C 1+α Λ < C, 0 < inf θ n sup θ n < We represent the free boundary Γ t in the form Xλ, t = X 0 λ + ht, λe n λ, 4.4 and denote by t the domain bounded by Γ t and G and by t the domain bounded by Γ t and S see Figure 1. Then 4.1 can be written as 20

21 X t λ, t = v n Xλ, t, t X λi λ, t v nxλ, t, t e i H i, 4.5 where H i = H i h, λ are the Lamé coefficients associated with the coordinate system [35. By 4.4, we have Xλ e n = λ i X 0 λ λ i + and the last term vanishes since e 2 nλ = 1. By 4.4, we also have Hence, 4.5 is equivalent to hλ, t t + hλ, t e n λ e n + hλ, t e nλ e n, λ i λ i X t λ, t e n λ = h t λ, t. v n Xλ, t, t e i H i hλ, t λ i = v n Xλ, t, t e n Using 4.2 one can show that e n λ, e i λ and H i are in C 3. We introduce a class of functions hλ, t by where h T = We need the following lemma X 0 λ vn Xλ, t, t e i e n 4.6 λ i H i W T,M = {hλ, t; hλ, 0 = 0, h T M}, M > 0, sup 0 t T h, t C 2+α Λ + sup Lemma 2 Consider the hyperbolic equation for w = wλ, t, λ Λ j=0 xhλ, j C α [0,T. w t + bλ, t λ w = Gλ, t, λ Λ, 0 t T, T > with initial condition and assume that, for an integer m 1, w t=0 = 0, 4.8 b, t C m+α Λ + G, t C m+α Λ K, t [0, T. Then there exists a unique solution of 4.7, 4.8 satisfying w, t C m+α Λ C 1 KT, 4.9 m sup xwλ, j C α [0,T C 1 KT 1 α, 4.10 λ Λ j=0 m 1 j=0 where C 1 K depends only on K. j xw t, t C1 Λ C 1 KT + K if m 1,

22 Proof. The proof can be carried out in a same way as in [5, Lemma 2.2, integrating along the characteristics to obtain 4.9, and making use of the special zero initial condition 4.8. Then, using the equation 4.7 and the estimates 4.9, we immediately obtain Next, using 4.7 and the estimates 4.11, we obtain the estimates 4.10 for j m 1. Finally, the estimate for j = m in 4.10 is obtained by estimating first in the direction of characteristics and then using 4.9, similarly to the argument in [5. For any h W T,M, we define X by 4.4. We know from Section 3 that the static reduced problem has a unique solution satisfying the Schauder estimates 3.47 with C = CM which depends on M = h T. In particular, for 0 t T, v n, t C 2+α t CM, 4.12 where CM depends on M = h T. We next extend the function v n x, t into t Γ t such that v n, t C 2+α t t c 0CM, c 0 > For simplicity, we denotes the extended function also by v n. We shall need to use the same extension procedure for any v n corresponding to any h in W T,M. We can achieve such an extension by first extending v n along each normal to the surface x = Xt, λ by a polynomial of degree two in the distance along this normal to achieve smooth C 2+α extension and then multiply this extension by a fixed C 3 cutoff function ψ which equals to 1 in t and vanishes outside a small neighborhood of t, independent of t, 0 t T T small. We define a function hλ, t by hλ, t t + hλ, 0 = 0, v n Xλ, t, t e i H i hλ, t λ i = v n Xλ, t, t e n X 0 λ vn Xλ, t, t e i e n, λ i H i where v n is the extension defined above. We note that the assumption 4.2 will be needed in order to prove that hλ, t is in C 2+α. We introduce the mapping S : h h. Clearly h is a fixed point of S if and only if the corresponding v n and Xt, λ form a solution of the reduced free boundary problem. Applying Lemma 2 to h, we get h, t C 2+α Λ C 1 MT, and hλ, Cα [0,T + x hλ, Cα [0,T + 2 x hλ, Cα [0,T C 2 MT 1 α ; here the C i M are constants depending only on M. Taking T such that C 1 MT + C 2 MT 1 α < M, we conclude that S maps W T,M into itself. We next show that S is a contraction, and hence, it has a unique fixed point in W T,M. Take h 1, h 2 W T,M and the corresponding functions X 1, v n, v s, P 1, P 2, h 1 and X 2, w n, w s, Q 1, Q 2, h 2, and set We use the transformation δ = h 1 h 2 T. θ = θ, φ = φ, r = r h1 λ, t h 2 λ, tψr, θ, φ in order to estimate the differences between the two solutions; here ψ is a cut off function which equals to 1 in a small neighborhood of the free boundary. By the same argument used in the proof of 4.12, we can estimate the Cx 2+α norm of v n w n in the transformed domain v n w n C 2+α x CM h 1 h 2 T CMδ

23 Using the equations for h i, and Lemma 2 after dividing by δ, we obtain, h 1 h 2, t C 2+α Λ + x h j 1 h 2 λ, j=0 Cα [0,T CMT + T 1 α δ < δ 2, 4.15 if T is small enough. Hence S is a contraction, and it has a unique fixed point. We summarize: Theorem 3 Consider the free boundary problem , 4.1. If the conditions 4.2, 4.3 hold then there exists a unique solution v 1, v 2, v 3, P 1, P 2, h, for a small time interval 0 t T, satisfying the estimates sup 0 t T v 1 C 2+α t + h, t C 2+α Λ + sup λ Λ j=1 xh j Cα [0,T <, 4.16 v j C 2+α t + P 1 C α t + P 2 C α t < 4.17 j=2 Remark 4 Since the C 3+α Schauder estimates 3.48 are also valid, existence can be obtained also in the framework of C 3+α estimates, assuming that X 0 C 4+α, θ n C 3+α, f j C 1+α, g j, h j C 2+α, i.e., the estimates in 4.17 hold with C 2+α replaced by C 3+α for v i, and C α replaced by C 1+α for P i. 5 Existence and uniqueness of solution for the full problem In this section we assume that f j C 1+α,α/2 E [0,T, f j C α,1+α/2 E [0,T < C, j = 1, 2, 3, g j C 2+α,α/2 E [0,T, g j C 1+α,1+α/2 E [0,T < C, j = 1, 2, h j C 2+α,α/2 E [0,T, h j C 1+α,1+α/2 E [0,T < C, j = 1, 2, G C 1+α,α/2 E [0,T < C 5.1 where E is a neighborhood of 0 0, and that the initial free boundary satisfies: X 0 = X0, C 4+α Λ. 5.2 In the previous sections 3 and 4, the function θ n was a given function. In this section, we shall solve the full free boundary problem by taking θ n to be coupled to v i and the free boundary defined in terms of hλ, t, where θ n satisfies the parabolic equation with boundary conditions and initial conditions θ n t + θ nv 1 = ε θ n + Gx, t in t, 0 t T ε > θ n n = 0 on t, 0 t T, 5.4 θ n x, 0 = θ 0 x for x, 0 < inf θ 0 x sup θ 0 x < We shall use the contraction fixed point theorem establish the existence of a unique solution for a small time interval 0 t T. We shall define a set W T,M1 of functions hλ, t, λ Λ, 0 t T, and a set Z T,M2 of vectors v 1, v 2, v 3. Then for any h old W T,M1 and v1 old, vold 2, vold 3 Z T,M 2, we shall solve the parabolic equation 5.3 with 5.4 and 5.5 to obtain θ new, and then apply Theorem 3 to the reduced free boundary problem with θn new, to obtain new functions v1 new, v2 new, v3 new and h new. We define a mapping S by Sh old, v old 1, v old 2, v old 3 = h new, v new 1, v new 2, v new 3,

24 and shall prove that S is a contraction mapping on W T,M1, Z T,M2, if T is sufficiently small. But in order to accomplish this, we need θ n to have sufficient regularity in t. We are unable to accomplish this if we take hλ, t to be in one of the standard Schauder spaces. Instead, we shall require the function hλ, t to have a rather unusual type of regularity in λ and t, namely, to belong to the class W T,M1 = {hλ, t; hλ, 0 = 0, h T 1, h t L + x h t L + 2 xh t L M 1 }, M 1 > 0, 5.7 where h T = sup 0 t T + h, t C 3+α Λ + sup xhλ, 3 Cα [0,T λ Λ 1 sup sup j=0 λ Λ 0<t 1<t 2<T j x[h t λ, t 1 h t λ, t 2 t 1 t 2 1+α ε1/2, 5.8 and ε 1 < α is a small positive constant. Notice that the last term is roughly a bound on x 3+α ε/2 t h. The reason for introducing ε 1 will become clear later on when we show that it gives rise to a small factor T ε1/4 in a crucial estimate for h. We introduce the sets N T = t [0,T t {t} and T = t [0,T t {t}, and the variables v n in N T and v s in N T T by v n x = v 2 x, x N T, We take v n and v s in the spaces { v1 x, x T v s x = v 3 x, x N T j xv n C α,α/2 N T, j = 0, 1, 2, j xv s C α,α/2 N T C α,α/2 T, j = 0, 1, 2, and we define the set Z T,M2 is given in terms of v n, v s by restricting them as follows: Z T,M2 = { vn, v s ; j xv n C α,α/2 N T M 2, j xv s C α,α/2 N T M 2, j xv s C α,α/2 T M 2, j = 0, 1, 2, } We also assume that θ 0 is in C 3+α θ 0 0, n = 0 on 0, 5.9 and the second order compatability condition is satisfied on 0 {t = 0}. This compatibility condition, on the free boundary x = Xλ, t at t = 0, is the condition that [ d θn lim Xλ, t, t = 0, t 0 dt n where θ n t is computed from the differential equation 5.3 and all the derivatives j xθ n are replaced, as t 0, by j xθ 0. Similarly, one defines the compatability condition on G {t = 0}. Choosing M 1. Under our assumption 5.8, the function h old is clearly C 3+α ε1,3+α ε1/2, with a bound depending only on M 1. From the theory for parabolic equations [26, θ n C 3+α ε1,3+α ε1/2. In particular, the parabolic theory asserts that xθ j n, j = 0, 1 are Lipschitz continuous in t, xθ 2 n is C 1+α ε1/2 Hölder continuous in t, and xθ 3 n is C α ε1/2 Hölder continuous in t. Here we have not used the full norm in 5.8, i.e., the bound on the term with j = 1 in 5.8 was not used. With the j = 1 term in 5.8, the solution θ n is more regular in spatial direction. After flattening the boundary locally as we did in section 3, the coefficients of the equations for θ n involve h old t and xh j old j = 0, 1, 2. Furthermore, we can choose the flattening map such that the homogeneous Neumann boundary condition remains intact. The j = 1 term in 5.8 makes it possible for differentiating in x-direction: when differentiating in 24