LONG-TIME EXISTENCE OF CLASSICAL SOLUTIONS TO A 1-D SWELLING GEL

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1 LONG-IME EXISENCE OF CLASSICAL SOLUIONS O A -D SWELLING GEL M. CARME CALDERER AND ROBIN MING CHEN Abstract. In this paper we derived a model which describes the swelling dynamics of a gel and study the system in onedimensional geometry with a free boundary. he governing equations are hyperbolic with a weakly dissipative source. Using a mass-lagrangian formulation, the free-boundary is transformed into a fixed-boundary. We prove the existence of long time C -solutions to the transformed fixed boundary problem.. Introduction. In this paper, we study existence of long time C -solutions of a free boundary problem modeling swelling of gels, in one space dimension. he gel is assumed to be a mathematical mixture of polymer and solvent. We assume that the gel is surrounded by solvent and that its boundary is fully permeable to solvent. he governing equations consist of the laws of balance of mass and linear momentum of the components and form a weakly dissipative hyperbolic system. he scaling leading to hyperbolic dynamics is motivated by the modeling of polysaccharide gels occurring in nature, for instance as motor devices of gliding bacteria myxobacteria. In general, hyperbolic dynamics is a signature property of active gels encountered in models of the cytoskeleton of living cells and also in studies of biomimetic non-equilibrium gels [4], [9]. We assume that a gel is a saturated, incompressible and immiscible mixture of polymer and solvent. Incompressibility in the context of mixtures refers to every component having constant mass density in the pure state. In particular, it does not preclude the polymer from experiencing deformations with very large change in volume, as much as 5 percent, if ions are present. Immiscibility corresponds to the constitutive equations depending on the volume fraction, φ φ and, respectively, of the polymer and solvent component. he saturation assumption also known in some literature as incompressibility is the statement of the volume fractions adding to. he balance laws consist of the equations of balance of mass and linear momentum of each component together with the saturation constraint. he coupling between component equations takes place through the Flory-Huggins energy of mixing, the drag forces and the boundary conditions. For gels made of entangled polymers such as those used in biomedical devices, scaling arguments show that elastic [2, 2, 23] and dissipative effects often dominate inertia, that enters the early dynamics only. Moreover, in [5] the authors assume that the dissipation of both the polymer and solvent is Newtonian. In this paper, we consider gels made of polysacharide networks, with dissipation parameters several orders of magnitude smaller as much as 7 than those of the entangled polymer counterparts [22]. his motivates us to keeping the inertia terms to study the intermediate time dynamics of the polysaccharide systems as well as the early dynamics of the highly dissipative gels. he energy of the system is the sum of the nonlinearly elastic stored energy function of the polymer and the Flory-Huggins energy of mixing. We establish a dissipation law satisfied by solutions of the governing system in arbitrary space dimension. here are two types of boundary conditions complementing the elasticity and the Euler equation. hese are of traction-displacement kind, for the gel equation this being the sum of the elasticity and the Euler equation for the fluid, and either scalar Dirichlet or traction boundary conditions for the the Euler equation of the fluid. hese boundary conditions, also considered in earlier works on gels [22], [6], are motivated by the works of Doi and Yamaue [2, 2]. hey express properties of permeability of the gel interface to the surrounding fluid, and may range from impermeable Dirichlet boundary to fully permeable traction boundary. However, these conditions have been found not be exact, from the point of view that they do not necessarily guarantee the balance of mass, linear momentum and energy at the interface between the gel and the external fluid. In fact, they differ from the balance laws at the interface by a term quadratic on the normal component of the difference of the polymer and solvent velocities, and proportional to the fluid inertia. Moreover, the assumption of viscous interface stress yields a fully dissipative energy law [3]. In this work, we consider a polymer separated from the surrounding fluid and confined to a strip domain. We assume that at time t = the polymer enters into contact with the fluid at the boundary points, causing it to enter and swell the polymer. he requirement for the three dimensional governing system to have onedimensional solutions determines the pressure Lagrange multiplier. Moreover, it follows from the equation of balance of mass, that the center of mass velocity is independent of space. his motivates us to rewrite the governing equations in terms of the relative fluid-polymer velocity, U = v v 2. Another implication of the one dimensional geometry is that the volume fraction on the boundary is fully determined from the traction -

2 boundary condition, provided the fluid inertia on the boundary conditions is neglected. In this case, a problem originally formulated with traction boundary becomes a of Dirichlet type. Without loss of generality and to simplify the analysis, we assume that the initial domain also used as reference configuration to be the symmetric interval,. We use the mass-lagrangian change of variables from gas dynamics, to transform the free boundary problem into one with fixed domain. We find that G φ + U 2 < is a necessary and sufficient condition for the governing equation to be hyperbolic, and so, it is necessary that G φ < hold. he function Gφ corresponds to a one-dimensional stress, and the condition of negative derivative with respect to φ expresses monotonicity of the stress with respect to the deformation gradient u x = φ I φ, where φ I denotes the reference volume fraction of the polymer. Heuristically, if G is bounded, the hyperbolicity condition is satisfied for sufficiently small relative velocity. Using the framework introduced in [2] in dealing with the hyperbolic initial-boundary-value problem, we obtain the local well-posedness of strong solutions for data satisfying the hyperbolicity and uniform Kreiss-Lopatinskiĭ condition. We also consider the long-time existence of classical solutions. We use the method of characteristics to estimate the C -norm of solutions. Due to the lack of strong dissipation of the system and the boundary damping, our estimate does not lead to the existence of global solutions. Instead, we prove that the existence time is at least of order O log ε, where ε is the C -deviation from the equilibrium. Such a small-data large-time existence has to be expected since large data are expected to form singularities. On the other hand, the hyperbolicity and the dissipative structure make the system amenable to the weak solution theory. In fact the Cauchy problem of the D system has been considered in [4], where an entropy-entropy flux pair was constructed, allowing one to obtain BV solutions. As for the free boundary problem, at the linearized level, the presence of shocks is precluded [4]. It would be interesting to know if one can use the entropy information to build up weak solution for large data in the free boundary setting, but that is beyond the scope of our present ambitions. he rest of the paper is organized as follows. In Section 2 we set up the mathematical model that describes the swelling dynamics of a gel and derive the constitutive equations. We also formulate the one-dimensional system with boundary conditions. In Section 3 we transform the one-dimensional free-boundary problem into a fixed-boundary problem using the mass-lagrangian change of variables, and prove the local well-posedness of strong solutions to the transformed problem. In Section 4 we prove the long-time well-posedness of classical solutions. 2. General model of gel-swelling. We begin by setting up the model of the hydrogel postulated in [4, 5]. We assume that a gel is a saturated, incompressible and immiscible mixture of elastic solid and fluid. In the reference configuration, the polymer occupies a domain Ω R 3. he solid undergoes a deformation according to the smooth map 2. x = xx, t, such that det x >, X Ω. We let = xω denote the domain occupied by the gel at time t, be the gel-solvent interface in the case that =, the gel is fully surrounded by the solvent, and let F = x denote the deformation gradient. We label the polymer and fluid components with indices and 2, respectively. A point x is occupied by, both, solid and fluid at volume fractions φ = φ x, t and = x, t, respectively. We let v = v x, t and v 2 = v 2 x, t denote the corresponding velocity fields. An immiscible mixture is such that the constitutive equations depend explicitly on the volume fractions φ i. We let ρ i denote the mass densities of the ith component per unit volume of gel. hey are related to the intrinsic densities γ i by ρ i = γ i φ i, i =, 2, which are constant in the case of an incompressible mixture. 2.. Governing equations. he assumption of saturation of the mixture, that is, that no species other than polymer and fluid are present, is expressed by the equation 2.2 φ + =. In some terminologies, this is condition is also known as incompressibility. he governing equations consist of the balance of mass and linear momentum of each components as well as the chain-rule relating the time derivative to the gradient of deformation with the velocity gradient: 2.3 φ i t + φ iv i =, 2-2

3 vi γ i φ i t + v i v i = i + f i, f + f 2 =, F t + v F = v F, x, i =, 2. Here f i represent drag forces between the components, and i is the Cauchy stress tensor of the ith component. Note that we have used the incompressibility assumption γ i = constant in the previous equations. he Lagrangian form of the equation of balance of mass of the polymer component is 2.6 φ det F = φ I in Ω, where φ I is the initial polymer volume fraction in the reference configuration. Note that if det F =, then φ I = φ, so that no changes in the volume fraction of the gel correspond to no changes in volume fraction. Hence equation 2.6 is equivalent to equation 2.3 with i =. Adding equations 2.3, and using the constraint 2.2 gives 2.7 φ v + v 2 =. With this formulation, the final system of the governing equations are 2.4 and 2.5, subject to the constraints in 2.2, 2.6, and Boundary conditions. We now specify boundary conditions at the gel-water interface. We observe that the equations of balance of mass are first order, scalar equations for φ and that require prescribing initial conditions only. Later, we will assume that the network part of the gel is elastic, with Newtonian dissipation. he latter will also be assumed for the in-gel and exterior fluids. his requires one vector boundary condition for each equation. here are two kinds of boundary conditions that we will discuss, in order to come up with a total of six scalar boundary conditions for the system: statements of balance of mass, linear momentum and energy across, and relations expressing the permeability properties of the interface. Balance of mass across. We denote v f the velocity field of the fluid outside the gel network, which is assumed to be Newtonian. he equation of balance of mass of the fluid across the interface is 2.8 v f v n = v 2 v n := w, where n denotes the unit normal to, pointing from the gel to the pure liquid. Note that the last relation defines the normal component, w, of the relative velocities. We also let t and t 2 denote a pair of orthonormal vectors perpendicular to n. Let v be a vector field on. We use the notation to denote the vector components tangent to at a point. hat is, we write v = v nn + v = v nn + v t i t i, i=,2 Continuity of the components of the velocity tangent to. We assume the following no-slip condition 2.9 v f v = v 2 v := q. Remark 2.. he no-slip assumption equating the tangential components of the fluid velocity, relative to the polymer one, across may admit a generalization of the form 2. v f v = η v v 2 v := q, where η v > is a dimensionless parameter related to the surface viscosity. We will observe next that this brings tangential stress components in the balance of linear momentum equation. Balance of linear Momentum across. We next consider force balance across the interface. Consider a point x at a time t, and let us observe this point in an inertial frame traveling with the same velocity as the polymer at point x and time t. Given equations 2.8 and 2.9 we observe that the water in the fluid region 2-3

4 travels at velocity wn + q and in the gel region travels with velocity w n + q. the change of mass of water across the interface per unit time at point x at time t is given by γ 2 w. As water comes out of or into the gel region, corresponding to the collapsing w > or swelling state w <, respectively, there is the following amount of momentum change from the gel region to the water region per unit time: γ 2 w wn w 2. n. he force acting at the interface is given by: n. he law of balance of linear momentum gives 2.3 Note that this implies, in particular, that + 2 n = γ 2 w w φ2 n n t i =, i =, 2. Note that the difference in the stress across the boundary is normal to the interface. In particular, we have: 2.5 n + 2 n n q = since q is tangential to the surface. At this point, we have the boundary conditions 2.8, 2.9 and 2.3 on, which together give us six boundary conditions. Mass balance and total force balance would provide the necessary number of boundary conditions if the interior of were composed of a one-phase medium. Here, the interior of is a two-phase gel. We thus require three additional boundary conditions, assuming that all of the phases have bulk viscous stresses. his corresponds to specifying some condition that involves w and q. he appropriate forms for these boundary conditions will be discussed shortly. We now check that the boundary conditions 2.8, 2.9 and 2.3 lead to mass and momentum conservation. Let us first check that the amount of water is conserved: d dx + dx dt Ω c t = v 2 v nds v f v nds =, where we used 2.8 in the second equality. Now, we turn to momentum conservation. We have: d γ φ v + γ 2 v 2 dx + γ 2 v f dx dt Ω c t = γ 2 v v 2 n v 2 + n + 2 n ds γ 2 v v f n v f + n ds = γ 2 v 2 v f w + n + 2 n n ds = γ 2 w w n + n + 2 n n ds = 2-4

5 where we used 2.8 in the second and third equalities, 2.9 in the third equality and 2.3 in the last equality. Balance of energy across. Next, we consider energy conservation. he final form of our energy relation will lead us to possible forms for the additional boundary conditions we shall impose on our system. d dt 2 γ φ v γ 2 v dx + Ω c 2 γ 2 v f 2 dx t = v + v 2 2 dx + v f dx Ω c t + 2 γ 2 v 2 2 v v 2 n 2 γ 2 v f 2 v v f n ds = v + v 2 2 dx v f dx Ω c t + n v + 2 n v 2 n v f ds + 2 γ 2 v 2 2 v v 2 n 2 γ 2 v f 2 v v f n ds. 2.7 Let us evaluate the last two boundary integrals. Using 2.8 and 2.9 we have: n v + 2 n v 2 n v f 2 = n + 2 n n v + n n = γ 2 w 2 v n + n 2 n n n w + 2 n n q n n w n q where we used 2.8 and 2.9 in the first equality and 2.5 in the second equality. On the other hand, γ 2 v 2 2 v v 2 n 2 γ 2 v f 2 v v f n = 2 γ 2 v + w n + q 2 w + 2 γ 2 v + wn + q 2 w =γ 2 w 2 2 w v n 2 γ 2 2 γ 2w 2 w where we used 2.8 in the first equality. We may go back to 2.6 to conclude that: d dt 2 γ φ v γ 2 v dx + Ω c 2 γ 2 v f 2 dx t = v + v 2 2 dx v f dx Ω c t n n 2 γ 2w 2 2 n n 2 w 2 γ 2 w ds n q ds. he last two boundary integrals denote the change in energy coming from the surface. We would like these terms to be negative. One way to achieve this would be to let: η w= n n 2 γ 2w 2 2 n n 2 w γ 2, 2.2 η q= n, 2-5

6 where η and η are positive constants and n denotes the component of n that is tangential to the membrane. he above conditions provide the additional three boundary conditions we need on. he boundary condition 2.2 depends quadratically on w and is physically reasonable only if w is sufficiently small. his difficulty will not arise if we neglect inertial terms and set γ i =, i =, 2. For most practical situations, inertial effects can be safely neglected. If we substitute 2.2 and 2.2 into 2.9, we have: d dt 2 γ φ v γ 2 v 2 2 dx + Ω c 2 γ 2 v f 2 dx t = v + v 2 2 dx v f dx η w 2 + η q 2 ds. Ω c t 2.3. Energy dissipation and constitutive equations. he free energy density Ψ of the gel consists of the elastic energy W P F of the polymer and the Flory Huggins energy of mixing W FH φ, [, ]: Ψ = φ W P F + W FH φ,, with W FH φ, = a φ log φ + b log + c φ, a = K B, b = K B, c = K B χ, V m N V m N 2 2V m where various parameters above are of the following physical interpretations:. K B is the Boltzmann constant, and is the absolute temperature; 2. V m is the volume occupied by one monomer; 3. N, N 2 are the numbers of lattice sites occupied by the polymer and the solvent, respectively; 4. χ = χφ, is the Flory interaction parameter; In this way, the total energy of the gel is given by 2.24 E= := E P + E S. γ 2 φ v 2 + γ 2 2 v Ψ dx + Ω c t γ 2 2 v f 2 dx. We now show that smooth solutions to the governing equations satisfy a dissipation inequality, and the dissipation inequality suggests the exact form of the Cauchy stress tensors for the ith component, i. We define i as the sum of reversible stress r i and viscous stress v i for i =, 2: 2.25 i = r i + v i. Specifically, the dissipation inequality allows us to obtain the exact form of the reversible stress r i, the viscous stress v i, and the expressions for the friction forces f i. he derivation of the dissipation inequality uses a similar approach as in [5], and we here follow the presentation from [3]. heorem 2.2. Dissipation Relation Suppose that {v i, φ i } are smooth solutions of equations with boundary conditions 2.8, 2.9, 2.3, 2.2 and 2.2 on. Let p denote the Lagrange multiplier corresponding to the constraint 2.7. Assume that the following constitutive equations for the stress tensors components hold: [ r W P = φ F F WFH φ W ] FH 2.26 W FH + pφ I, φ r 2 = pi, v i = η i Dv i + µ i v i I with i =, 2, where η i > and µ i > denote the shear and bulk viscosities of the ith component, respectively, and Dv := 2 v + v is the symmetric part of the velocity gradient. Suppose that the solvent-polymer friction forces are given by 2.29 f = p φ κv v 2, f 2 = f, 2-6

7 with κ > denoting the gel permeability coefficient. hen the following dissipation relation holds: [ ] de dt = ηi Dv i 2 + µ i v i 2 + κ v v 2 2 dx i 2.3 η w 2 + η q 2 ds where n represents the unit outer normal to. Proof. Without loss of generality, assume that γ = γ 2 =. We observe that the boundary of the gel is determined by that of the polymer network and moves with velocity v. We first calculate the time derivative of E. Using the Reynolds ransport heorem, and taking into account that the boundary moves with the network speed v, we have [ de P φ = dt t 2 v 2 + φ ] 2 2 v 2 2 φ + v 2 v 2 + v 2 v 2 2 dx [ ] Ψ + t + v φ W P + v W FH dx. Next, expand Ψ t de P dt in terms of φ,, F using the chain rule to obtain [ φ t 2 v 2 + φ 2 2 v 2 2 φ + v 2 v 2 + v [ + W P + W ] FH WFH W P φ,t +,t + φ φ F : F t dx [ ] + v φ W P + v φ W P + v W FH dx. = Equation 2.3 of balance of mass, φi t [ de P φ = dt t 2 v 2 + [ + W P + W FH φ + = φ i v i, for i =, 2 yields 2 v 2 2 φ + v φ v 2 v 2 WFH [ ] v φ W P + v φ W P + v W FH dx. ] 2 v 2 dx ] + v 2 v 2 2 dx ] W P v 2 + φ F Gathering the terms involving W FH and using the incompressibility of mixture constraint in equation 2.2 gives the following: W FH φ v W FH v 2 + v W FH + W FH v φ φ 2 = φ v W FH + W FH WFH + v φ + W FH + W FH v φ φ [ WFH = φ W ] FH + W FH v φ Substituting it into the expression for de P dt [ de P dt = + t gives φ 2 v v 2 2 { φ W P F : F t + φ + v [ WFH φ W FH φ : F t ] 2 v 2 + v 2 v 2 2 dx ] } + W FH v + φ v W P dx dx. 2-7

8 2.3 Next, we apply the chain-rule relation between material and spatial time derivatives, to obtain the following identity: φ W P F DF dt = F t + v F = v F, : F t + φ v W P WP F = φ W P F : F t + φ v = φ W P F he expression for de P dt simplifies to [ de P φ = dt t 2 v v 2 2 { W P + φ F : v F + : F : F t + v F = φ W P F : v F. φ + v 2 v 2 [ WFH φ W FH φ + v ] 2 v 2 2 } + W FH ] v In the final steps, we address the kinetic energy terms. Using the equations of balance of mass and linear momentum of for i =, we calculate: t 2 φ v 2 + v 2 φ v 2 = φ v v,t + 2 v 2 φ t + φ v + 2 φ bv v 2 = φ v v,t 2 v 2 φ v + 2 v 2 φ v + φ v v v = φ v Dv Dt = v + f D Here, Dt denotes the material time derivative. Likewise, t 2 v v 2 v 2 2 = v 2 v 2,t + 2 v 2 2 φ2 t + v + 2 v v 2 2 = v 2 v 2,t 2 v 2 2 v v 2 2 v + v v 2 v 2 = v 2 v 2,t + v 2 v 2 v 2 + v v 2 v 2 v 2 v 2 v v 2 2 v v 2 = v 2 Dv 2 Dt + v 2 v 2 v v v 2 2 v v 2. Integrating the last term by parts gives [ ] t 2 v v 2 v 2 2 dx [ = v 2 Dv 2 Dt + v 2 v 2 v v 2 + ] 2 v 2 2 v v 2 dx = v 2 Dv 2 Dt dx + 2 v 2 2 v v 2 n ds. dx dx 2-8

9 Plugging in the Lagrange multiplier we have that, for i =, 2, de P = v i i + f i v i + p φ i v i dx dt i { [ W P + φ F : v WFH F + φ W ] } FH + W FH v ] dx φ + v 2 2 v v 2 n ds. 2 Integrating by parts on the term i for i =, 2 gives, de P dt = v i i pφ i v i + f i v i p φ i v i dx i { [ W P + φ F : v WFH F + φ W ] } FH + W FH v dx φ [ + i v i + ] v 2 2 v v 2 n ds he time derivative de S dt de S dt is treated in a similar way giving = v f dx v f + v f 2 v v f n ds. Ω c t 2 herefore the conclusion follows by adding up 2.32 and 2.33, and using i as in 2.26 and 2.28, f i as given by 2.29, and all the boundary conditions. We further assume that the polymer is an isotropic elastic material, that is, 2.34 W P F = W P I, I 2, I 3, I = tr C, I 2 = 2 [ tr 2 C tr C 2], I 3 = det C, where C = F F. Here we take W P F to be of the following form 2.35 W P F = I s c + α I r β I β I q 2 3 where c, α, β, β >, r and s, q >. As for the Flory-Huggins mixture energy, we adopt the interaction equation from Horkay et al. [3] as follows χφ, = χ + χ φ + χ 2. Further computation gives the following results for the reversible stress tensors r = 2sI s F F + [ α rdet F r + β det F + β qdet F q] I { [ KB φ χ + K B log φ + K B + 2χ φ + 3χ 2 2V m N V m N V m KB χ φ + K B log + K ] B + χ + χ 2 φ 3 2V m N 2 V m N 2 V m KB χ φ + K B φ log φ + K B log 2V m N V m N 2 V m r 2 = p I. +χ + χ 2 + pφ }I, 2-9

10 2.4. A new field of unknowns. Now we will formulate the effective governing equations in terms of the center of mass velocity V = φ v + v 2 and the diffusion velocity U = v v 2. Hence the new field of unknowns is he total stress of the system is {V, U, φ, p, F }. = r + r 2 φ U U, In this way, in Eulerian coordinates, the governing equations become φ 2.39 t + [V + φ U ] φ + φ [V + φ U] =, 2.4 V + V V =, t U t + 2φ UU U U φ + VU + UV 2.4 = β 2 φ φ φ φ U + λ φ φ φ, 2.42 F t + [V + φ U] F = [V + φ U] F, 2.43 V =. he first and second equations give the balance of mass and linear momentum for the mixture. he third equation can be interpreted as giving the evolution of the microstructure of the gel. Equation 2.42 is a version of the chain rule relating time derivatives of F with velocity gradients. his equation is required in mixed solid-fluid systems [8]. he last equation looks like an incompressibility condition of the mixture and is from the balance of mass and the constraint φ + =. In application to modeling gliding behavior of bacteria by polysaccharide swelling, the problem can be thought of as in one dimensional [2]. We consider the gel occupying a strip domain {x, y, z : L x L} for some L >. Now the fields of problem become 2.44 V = vx, t,,, U = ux, t,,, φ = φx, t, p = px, y, z, t. he deformation gradient is 2.45 F = diagdetf x, t,,. From 2.6, φx, t det F x, t = φ I. We can also write down the stress tensors 2.37 and 2.38 in the one-dimension formulation { = s Iφ s diag φ 2 I φ 2,, } α φ r I rφ r φ I I + β φ I + β qφ q I φ q I { [ KB φ χ φ + K B log φ + K B + 2χ φ φ + 3χ 2 φ 2V m N V m N V m KB χ φ + K B log φ + K ] B χ + χ 2 φ 3 2V m N 2 V m N 2 V m KB χ φ φ + K B φ log φ + K B φ log φ 2V m N V m N 2 V m } +χ φ + χ 2 φ + pφ I, = p φi. he second and third equations of 2.4 indicate that p = px, t is independent of y and z. Moreover, equation 2.43 together with the first component of equation 2.44 gives v = vt. Prescribing the initial 2-

11 condition v = leads to vt = for all t >, provided that = holds. he latter determines p in terms of φ and u, up to a constant. In this way we arrive at the following system for φ and u 2.48 where 2.49 φ t + [φ φu] x =, Gφ = K B log φ K B logφ N 2 V m N V m s φ2 I 2sφ 2 I I + 2φ2 [ ] u t + 2 u2 2φ Gφ = βu x φ φ, + q β φ q I φ q + rα φ r I φ r + K B χ φ V m 2χ φ + 3χ χ 2 + 4χ 2 φ Boundary conditions. We assume initially, the polymer occupies the domain Ω = {x, y, z : L < x < L}, and the solvent is in x > L. At a later time t >, the gel occupies the region = {x, y, z : S t < x < S 2 t}, where x = S,2 t are the positions of the interface between the gel and the pure solvent. herefore inside the polymer region S t < x < S 2 t equations 2.48 hold for φ and u. he general boundary conditions are described in Section 2.2. Here we further simply the problem by ignoring the inertial effects from the system and hence the stress balance 2.3 now becomes n = n, on. he other boundary conditions we impose here for our problem characterize the degree of permeability of the interface see also, for instance, [9, 2, 2]. Again with no inertial effects, 2.2 becomes η w = n n n n. We now classify the boundary permeability in the following way he interface is fully permeable if η =. hus n n n n =, on. 2 he interface is impermeable if η =. In this case w =, that is 2.53 v n = v 2 n, on. 3 he interface is semipermeable for η,. In this paper we will consider the fully permeable interface. Hence from 2.5 and 2.52 we know that n φ 2 n =. In one dimension, plugging in 2.46, 2.47 and the above we obtain { = s Iφ } s φ 2 I φ 2 α φ r I rφ r φ I + β φ + β qφ q I φ q { [ KB φ χ φ + K B log φ + K B + 2χ φ φ + 3χ 2 φ 2V m N V m N V m KB χ φ + K B log φ + K ] B χ + χ 2 φ 3 2V m N 2 V m N 2 V m KB χ φ φ + K B φ log φ + K B φ log φ 2V m N V m N 2 V m } +χ φ + χ 2 φ. on. 2-

12 herefore we can determine the saturation value φ = φ at the interface. In other words, the locations of the interface x = S t and x = S 2 t are determined by the saturation value φ, i.e. φt, S t = φt, S 2 t = φ. he kinematic boundary condition asserts that the interface moves with the speed of the polymer, which means S t = [ φt, S t]ut, S t and S 2t = φt, S 2 tus 2 t, t. herefore we have obtained the initial and boundary conditions as follows φx, t = φ, at x = S t, S 2 t S = L, S t = [ φt, S t]ut, S t 2.55 S 2 = L, S 2t = [ φt, S 2 t]ut, S 2 t φx, = φ, ux, = u, for L < x < L. 3. he transformed problem. In this section we aim to setup a fixed-boundary problem associated to 2.48 and 2.55 and establish the local-wellposedness of strong solutions. o transform the free boundary to a fixed boundary, we perform the following change of coordinates 3. y = x S t φz, t dz, τ = t. Because S 2t S t φ dz gives the total mass of the polymer, we may normalize that to be. In this way the free domain S t, S 2 t becomes the fixed domain,. herefore the free boundary problem now turns into 3.2 φ τ + φu y φ y u =, u τ uu y u 2 φφ y G φφφ y = φy, τ = φ, for y =, φy, = φ, uy, = u, for < y <. βu φ φ, o further rewrite the system, we let ψ = /φ and fs satisfy that f s = sg s. hen let F s = f/s. In this way the above system becomes the following initial-boundary value problem ψ u + ψ u τ u2 2ψ 2 F ψ = βuψ, in,, ψ y 3.3 ψ = ψ, at y =,, ψ u τ= ψ = u, for < y <, where ψ = /φ and ψ = /φ. he gradient matrix is u ψ 2 Aψ, u = with eigenvalues 3.4 λ,2 ψ, u = and the corresponding left and right eigenvectors are 3.5 L,2 ψ, u = ψ ψ u 2 + G /ψ ψ 3 u ψ 2 [ ] u u 2 + G /ψ ψ ψ 2, ψ u 2 + G /ψ, ψ 3-2

13 and 3.6 R,2 ψ, u = ψ ψ u 2 + G /ψ. In the range of physical parameters corresponding to semi-dry polymer we have < φ <, hence < ψ. hus the system 3.3 is hyperbolic if 3.7 u 2 + G /ψ <. Hence G φ < will be needed to guarantee hyperbolicity of the governing system, and therefore is a requirement for the propagation of the swelling front towards the solvent region. It turns out that this condition is satisfied for polymer data see Fig.. However in the case of polysaccharide data, there may be multiple quantities φ c such that G φ c = see Fig.2. his may be interpreted in terms of the onset of deswelling, observed in bacteria motility phenomenon [2]; it may also be associated with volume phase transitions observed in systems with a small elastic shear modulus [6]. Gφ G φ φ φ one dimensional stress Gφ graph of G φ Fig. 3.. G and G for polymer data Gφ G φ φ φ graph of G φ one dimensional stress Gφ Fig G and G for polysaccharide data parameter N N 2 q s r α β φ I χ χ χ 2 polymer N polysaccharide parameter values In concern with the local-wellposedness of system 3.3, we need to check the following conditions see [2]: C. Non-characteristic condition. he matrix Aψ, u is non-singular for ψ, u in a certain proper domain M. 3-3

14 C2. Normality. he boundary matrix B is of constant, maximal rank and R 2 = kerb E s Aψ, u at y = = kerb E u Aψ, u at y =, where E s Aψ, u is the stable subspace of Aψ, u and E u Aψ, u is the unstable subspace of Aψ, u. Note that in our problem, the boundary matrix is B = C3. Uniform Kreiss-Lopatinskiĭ UKL condition. for all ψ, u M there exists C > so that. 3.8 V C BV for all V in the unstable subspace of A ψ, u at y = and for all V in the stable subspace of A ψ, u at y =. Consider now our problem 3.3, the first two conditions C and C2 are equivalent to 3.9 λ ψ, u < < λ 2 ψ, u, which is, from u 2 < ψ ψ G /ψ. Notice that since ψ >, 3. implies that G /ψ < and moreover the hyperbolicity condition 3.7. As for the third condition C3, the stable and unstable subspaces of A ψ, u is spanned by R and R 2 as defined in 3.6 respectively. Hence the UKL condition 3.8 is satisfied when there exists a γ > such that 3. ψ ψ u 2 + G /ψ γ. In order to establish the wellposedness of the initial-boundary value problem 3.3 in some strong Sobolev space H m with m > 2 being some integer, the data should satisfy the compatibility condition p t ψ y, = p t ψy,, at y =,, for all p {,,..., m }. Since ψ is some constant, the above condition is simply { ψ = ψ = ψ 3.2 p t ψy, =, at y =,, for all p {,..., m }. We can now state our local-wellposedness result. heorem 3.. [2] If m > 2 is an integer, then for all ψ, u H m+/2 [, ] H m+/2 [, ] satisfying 3., 3., and the compatibility condition 3.2, there exists > such that the problem 3.3 admits a unique solution u H m [, ] [, ]. 4. Long time existence of classical solutions. he Cauchy problem of system 2.48 was discussed in [4]. he authors showed that the Cauchy problem is L -stable, and the source term is merely weakly dissipative. However, due to the result in [7], the global existence of BV solutions can still be obtained. In concern of classical C solutions to the initial-boundary value problem of 3.3, the short time existence can be established using the general approach introduced in Chapter 4 of [7]. o obtain large time of global C solutions, one needs to control the C -norm of solutions. It turns out that if the systems exhibits strong enough dissipation or boundary damping then solutions can be extended globally in time see, for instance [5]. However in our case, the above two types of damping are both weak. What we obtain is the large-time 4-4

15 existence and uniqueness of C solutions to the system 3.3, provided that the data are close enough the equilibrium state ψ,. Let η = ψ ψ. o simplify the notation we still choose x, t as the space-time variable. hen 3.3 becomes η η + Aη, u + P η, u =, in,, u u 4. η B u t = x, at x =,, η u t= η = u ψ = ψ u, for < x <, where Aη, u = u η + ψ η + ψ 2 η + ψ u 2 + G /η + ψ u η + ψ 3 η + ψ 2, P η, u = βuη + ψ 2 η + ψ. Now we state our main result. heorem 4.. Suppose that ψ satisfies 4.2 ψ ψ G /ψ > and the C -compatibility conditions of the initial and boundary data η = η =, 4.3 u ψ 2 η x + ψ ψ u x =. hold. hen for any >, there exists an ε > so that if η, u C ε, x, then system 4. admits a unique classical C solution for t [,. Moreover the estimate of the size parameter ε in terms of the final time can be formulated as ε Ce C, where C > depends on ψ. On the other hand, this gives a small-data long-time existence of the system 4., and the dependence of the existence time on the size ε of initial data can be estimated as O log ε. Proof. As is pointed at the beginning of this section, the local-in-time wellposedness of C solutions can be proved using the idea from [7]. For the time being, suppose that on the existence domain of C solution ηx, t, ux, t we have 4.4 η, ux, t ε, where ε > is a suitably small number so that 3. is satisfied for η, u ε. Because of condition 4.2 and continuity, such an ε exists. o get the large-time existence of solutions, it suffices to prove that we can choose ε > small enough so that for any fixed ε with < ε ε, there exists some δ = δε > small such that if 4.5 η, u C δ, x 4-5

16 holds, then on the whole existence domain of the C solution η, u we have 4.6 η, u, t C ε, t. First we perform a diagonalization. Let { vi = L i η, uη, u i =, 2, w i = L i η, u x η, x u i =, 2, where L i is the i-th left eigenvector defined in 3.5 with ψ replaced by η + ψ. It is not hard to see that v = v, v 2 and w = w, w 2 satisfy the following system of diagonal form cf. Chapter 3 in [5] 4.7 for i =, 2, where t v i + λ i x v i + κv + v 2 = t w i + λ i x w i + κw + w 2 = 2 j,k= c ijk v j v k + 2 j,k= κ = βψ 2 2ψ >, c ijk w j v k + 2 j,k= d ijk v j w k 2 j,k= d ijk v j w k c ijk, d ijk, c ikj and d ijk are continuous functions of η, u, λ i t, x = λ i η, u cf. 3.4 with ψ replaced by η + ψ. Denote the quadratic parts by Q i v, w = Q i v, w = 2 j,k= 2 j,k= c ijk v j v k + c ijk w j v k + 2 j,k= 2 j,k= d ijk v j w k i =, 2, d ijk v j w k i =, 2. he initial condition for v and w is obviously { vi t= = v i = L iη, u η, u i =, 2, w i t= = w i = L iη, u x η, x u i =, 2. Boundary condition for v: hus At x =, : L B L 2 v =. 4.8 At x =, : v = v 2. Differentiating with respect to t, we get the boundary condition for w: [ η η = B = B Aη, u u u t x L w = BAη, u L 2 ] P η, u = BAη, u w 2. η u x herefore 4.9 At x =, : w = w

17 herefore in order to prove 4.6, we only need to show that the same estimate holds for v, w C : 4. v, wx, t ε. provided that the initial data is small: 4. v, w δ, x. From 4.4, we may also assume that on the whole existence domain of the C solution v, w, 4.2 v, wx, t ε. Let λ min = min{ λ i t, x : t, x, i =, 2}, λ max = max{ λ i t, x : t, x, i =, 2}, = /λ max, 2 = /λ min. Since ε is chosen so that the hyperbolicity condition 3. holds, we see that λ max λ min > and hence and 2 are well-defined. By continuity we know that we can certainly pick some δ > small such that 4. holds on some time domain. Hence to prove 4., it is only necessary to show that there exists some ε > so small that for any fixed >, if 4. holds on the domain D = {t, x : t, x }, then it still holds on D +, provided that the C solution v, w exists on such domain. For this purpose, let 4.3 V i t = max x v ix, t, U t = max{v t, V 2 t}, W i t = max x w ix, t, i =, 2, U 2 t = max{w t, W 2 t}. Suppose that the C solution v, w exists on D + and let ξ = f i τ; x, t be the i-th characteristic passing through a point x, t D + with t +. hen 4.4 d dτ f iτ; x, t = λ i τ, f i τ; x, t, τ = t : f i t; x, t = x. From 3.9 we know that for v x, t there are two possibilities: he first characteristic ξ = f τ; x, t intersects the interval [, ] on x-axis with the intersection point f ; x, t,, see Fig. 3. t x, t x Fig

18 4.5 Integrating the first equation in 4.7 along the first characteristic ξ = f τ; x, t we get v x, t = e κt v f ; x, t From 4. and since 4. holds on D, we have 4.6 e κt v x, t δ + e κ ε + Cε 2 + δ + e κ ε + Cε 2 + κe κt τ v 2 f τ; x, t, τdτ e κt τ Q v, wf τ; x, t, τdτ. κe κτ V 2 τdτ + C κe κτ U τdτ + C 2 e κτ i= V 2 i τ + W 2 i τdτ e κτ U 2 τ + U 2 2 τdτ, where V i, W i and U i were defined in he first characteristic ξ = f τ; x, t intersects the boundary x = at point, τ x, t where τ x, t satisfies Obviously we have f τ x, t; x, t = t τ x, t. In case 2, for the second characteristic ξ = f 2 τ;, τ x, t passing through, τ x, t, there are still two possibilities: 2a his second characteristic intersects the interval [, ] on the x-axis with the intersection point f 2 ;, τ x, t,, see Fig. 4a. t x, t t x, t, τ x, t, τ x, t x a, τ 2 x, t x b Fig b his second characteristic intersects the boundary x = at the intersection point, τ 2 x, t see Fig. 4b, where τ 2 x, t = τ 2 τ x, t,, in which, τ 2 x, t stands for the t-coordinate of the intersection point of the second characteristic ξ = f 2 τ; x, t passing through a point x, t with the boundary x = : We have f 2 τ 2 x, t; x, t = t τ 2 x, t. 4-8

19 herefore 2 τ x, t τ 2 x, t. Noting 4.7 we get 2 2 t τ 2 x, t. herefore for t +, τ 2 x, t, τ 2 x, t 2 2. In case 2a, using boundary condition 4.8 we get e κt v x, t= e κτ v, τ = e κτ v 2, τ τ κe κτ v 2 f τ; x, t, τdτ τ κe κτ v 2 f τ; x, t, τdτ τ e κτ Q v, wf τ; x, t, τdτ τ e κτ Q v, wf τ; x, t, τdτ. Integrating the second equation in 4.7 along the second characteristic we have e κτ v 2, τ = v 2f 2 ;, τ Combing the above equalities we get e κt v x, t = v 2f 2 ;, τ τ τ τ κe κτ v f 2 τ;, τ, τdτ e κτ Q 2 v, wf 2 τ;, τ, τdτ. κe κτ v f 2 τ;, τ, τdτ τ e κτ Q 2 v, wf 2 τ;, τ, τdτ τ κe κτ v 2 f τ; x, t, τdτ τ e κτ Q v, wf τ; x, t, τdτ. We have two cases: i τ. hen it is easy to see that we have the same estimate as 4.6. ii τ >. hen e κt v x, t δ + e κ ε + Cε 2 + +C which is again, the same as e κτ i= τ V 2 i τ + W 2 i τdτ δ + e κ ε + Cε 2 + In case 2b, using the similar idea we obtain e κt v x, t= e κτ v, τ = e κτ v 2, τ κe κτ V τdτ + κe κτ V 2 τdτ τ κe κτ U τdτ + C τ κe κτ v 2 f τ; x, t, τdτ τ κe κτ v 2 f τ; x, t, τdτ τ e κτ U 2 τ + U 2 2 τdτ, τ e κτ Q v, wf τ; x, t, τdτ τ e κτ Q v, wf τ; x, t, τdτ. = e κτ2 v 2, τ 2 κe κτ v f 2 τ;, τ, τdτ κe κτ v 2 f τ; x, t, τdτ τ 2 τ τ e κτ Q 2 v, wf 2 τ;, τ, τdτ e κτ Q v, wf τ; x, t, τdτ. τ 2 τ 4-9

20 From 4.9 we know that τ 2. Hence we can again split case 2b in the following two possibilities: i τ. hen the estimate for v x, t is 2 e κt v x, t δ + e κ ε + Cε 2 + κe κτ V 2 τdτ + C e κτ Vi 2 τ + Wi 2 τdτ ii τ >. hen δ + e κ ε + Cε 2 + e κt v x, t δ + e κ ε + Cε 2 + Hence we always have +C 2 e κτ i= τ V 2 i τ + W 2 i τdτ δ + e κ ε + Cε 2 + e κt v x, t δ + e κ ε + Cε 2 + κe κτ U τdτ + C i= e κτ U 2 τ + U 2 2 τdτ. κe κτ V τdτ + κe κτ V 2 τdτ τ κe κτ U τdτ + C κe κτ U τdτ + C e κτ U 2 τ + U 2 2 τdτ. e κτ U 2 τ + U 2 2 τdτ. In the same way, we obtain similar estimates for v 2, w and w 2. herefore we have e κt U x, t δ + e κ ε + Cε 2 + κeκτ U τdτ + C eκτ U 2 τ + U2 2 τdτ, Now let e κt U 2 x, t δ + e κ ε + Cε 2 + κeκτ U 2 τdτ + C eκτ U 2 τ + U 2 2 τdτ. e κt Y x, t = δ + e κ ε + Cε 2 + κeκτ Y τdτ + C eκτ Y 2 τ + Y 2 2 τdτ, e κt Y 2 x, t = δ + e κ ε + Cε 2 + κeκτ Y 2 τdτ + C eκτ Y 2 τ + Y2 2 τdτ. hen obviously, U i Y i for i =, 2 and Y i satisfies d dt Y it = CY 2 + Y2 2, herefore t [, + ], Y i = e κ [ δ + e κ ε + Cε 2 ]. Y t e κ [ δ + e κ ε + Cε 2] + C Y τ 2 dτ, where Y t = Y t, Y 2 t. More precisely, we have t [, + ], [ ] e κ δ + e κ ε + Cε Y t [. Ce κ δ + e κ ε + Cε ]t 2 herefore for any given >, we can choose < ε < /2C sufficiently small and independent of. For any fixed ε < ε ε, there exists δ = δε > also independent of, such that 4.29 e κ δ + e κ /2ε Y t [ ε, t [, + ]. C e κ δ + e κ /2ε ] herefore we may choose ε so small that 2Cε < and C ε < /4e κ. hen choose δ ε/4. In this way, we further have log ε. his proves the theorem. 5-2

21 5. Conclusion. We studied the free boundary problem of swelling of a gel with inertia and elastic effects dominating the viscous ones, in one space dimension. his corresponds to early dynamics regime for many polymeric gels and also describes the main dynamics for polyssaccharide gel networks. We transform the free boundary problem of the weakly dissipative hyperbolic governing equation to one fixed domain. As a result of requiring the system to admit one dimensional solutions, it turns out that the polymer volume fraction at the interface with the surrounding fluid is fully determined, if we neglect fluid inertia in the boundary conditions. We find necessary and sufficient conditions for the system to be hyperbolic, this requiring a monotonicity property of a one-dimensional stress. In current work, we are studying the case that such condition fails, and found that it may be indicative of two possible phenomena, one being de-swelling, that is the interface will move backwards, and the second one may be related to the gel locally experiencing a volume phase transition. his is a phenomenon analogous to the volume transition in polyelectrolyte gels that occur when a relevant ion concentration reaches a critical value. Acknowledgement. M.C. Carme was supported in part by NSF grants DMS R.M. Chen was supported in part by NSF grants DMS We thank the referee for valuable comments. REFERENCES [] S.S. Antman, Nonlinear problems of elasticity, Springer, New York, 25. [2] Sylvie Benzoni-Gavage and Denis Serre, Multi-dimensional hyperbolic partial differential equations: First-order systems and applications, Oxford, New York, Clarendon Press, 27. [3] M.C. Calderer, H. Chen, C. Micek and Y. Mori, A Dynamic Model of Polyelectrolite Gels, preprint. [4] M.C. Calderer and H. Zhang, Incipient dynamics of swelling of gels, SIAM J. Appl. Math., 68 28, [5] M.C. Calderer, B. Chabaud, S. Lyu and H. Zhang, Modeling approaches to the dynamics of hydrogel swelling, Journal of Computational and heoretical Nanoscience, -74, 2. [6] B. Chabaud, Models, analysis and numerics of gels, Univeristy of Minnesota, Ph.D thesis, 29. [7] C.M. Dafermos, Hyperbolic systems of balance laws with weak dissipation, Journal of Hyperbolic Differential Equations, 3 26, [8] C.M. Dafermos, Hyperbolic conservation laws in continuum physiscs, Springer, 25. [9] M. Doi, Dynamics and Patterns in Complex Fluids, Springer, 99. [] P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Ithaca, New York, 953. [] D. R. Gaskell, Introduction to the hermodynamics of Materials, aylor & Francis, Washington, D.C., 995. [2] E. Hoiczyk, Gliding motility in cyanobacteria: Observations and possible explanations, Arch. Micro., 74 2, 7. [3] F. Horkay, A. Hecht, and E. Geissler, Fine structure of polymer networks as revealed by solvent swelling Macromolecules, 3 998, [4] A. Levine and F.C. MacKintosh he mechanics and uctuation spectrum of active gels J. Phys. Chem. B, 3 29, [5] a sien Li, Global classical solutions for quasilinear hyperbolic systems, volume 32 of RAM: Research in Applied Mathematics, Masson, Paris, 994. [6] Y. Li and. anaka, Phase transitions of gels, Annu. Rev. Mater. Sci., , pp [7] a sien Li and Wen Ci Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University, Durham, NC, 985. [8] F. Lin, C. Liu, and P. Zhang, On hydrodynamics of viscoelastic fluids, Comm. Pure Appl. Math., 58 25, pp [9] V. Sharma, A. Jaishankar, Y.C. Wang and G. H. McKinley, Rheology of globular proteins: apparent yield stress, high shear rate viscosity and interfacial viscoelasticity of bovine serum albumin solutions, Soft Matter, 7, 2, p [2]. Yamaue and M. Doi, he stress diffusion coupling in the swelling dynamics of cylindrical gels, J. Chem. Phys., 22 25, p [2]. Yamaue and M. Doi, heory of one-dimensional swelling dynamics of polymer gels under mechanical constraint, Phys. Rev. E, 69 24, p [22] H. Zhang, Static and dynamical configurations of gels, University of Minnesota,Ph.D thesis, 27. [23] J. Zhang, X. Zhao, Z. Suo and H. Jiang, A finite element method for transient analysis of concurrent large deformation and mass transport in gels, J. Appl. Phys, 5 29,