A dynamic approach to heterogeneous elastic wires

We consider closed planar curves with fixed length and arbitrary winding number whose elastic energy depends on an additional density variable and a spontaneous curvature. Working with the inclination angle, the associated $L^2$-gradient flow is a nonlocal quasilinear coupled parabolic system of second order. We show local well-posedness, global existence of solutions, and full convergence of the flow for initial data in a weak regularity class.


Introduction and main results
In this work we consider a sufficiently regular closed planar curve γ with density ρ describing a heterogeneous elastic wire. The density ρ can be used to model a distribution of mass or temperature, for instance. We assume that the bending stiffness of the wire depends on the density and consider the energy with κ the curvature of γ, the arc-length element ds = |∂ x γ(x)| dx and ∂ s = 1 |∂xγ| ∂ x . Here µ > 0 models the diffusivity of the density, the smooth positive function β : R → R is the density-modulated stiffness and c 0 ∈ R gives the spontaneous curvature. This energy is invariant under orientation-preserving reparametrizations. The first part of the energy can be seen as a Helfrich-type energy for curves with modulated stiffness. The second part, the Dirichlet energy of ρ, ensures control of the density. Modulo isometries of R 2 , the planar curve γ (parametrized by arc-length) together with its orientation is uniquely determined by its inclination angle. If the curve has length L > 0, the energy can be expressed in terms of an inclination angle θ : [0, L] → R and the density ρ : [0, L] → R by E µ (θ, ρ) = 1 2 L 0 β(ρ)(∂ s θ − c 0 ) 2 + µ (∂ s ρ) 2 ds, (1.2) using that κ = ∂ s θ. More precisely, we have E µ (γ, ρ) = E µ (θ, ρ), where θ is an inclination angle for γ. Note that in (1.2), the differential operator ∂ s is just the ordinary derivative with respect to s, the arc-length parameter. Similarly, ds just denotes integration with respect to the variable s ∈ for some ω ∈ Z, the rotation index of the curve. Notice that allowing ω to be negative we keep track of the orientation.
In the case of zero spontaneous curvature and fixed rotation index equal to one, this energy has previously been considered in [4], where the minimization problem under the constraints of fixed length and fixed total mass (of the density) is studied. Especially in the case c 0 = 0 it makes sense to fix the length of the curve, as otherwise a circle with suitable radius and a constant density is the trivial minimizer. The goal of this article is to introduce a dynamic approach to minimize the elastic energy E µ by evolving (θ, ρ) by the associated L 2 -gradient flow. In order to guarantee that θ describes a closed curve and to preserve the total mass of ρ, the evolution equations include nonlocal Lagrange multipliers. This results in the initial boundary value problem   for all t ∈ (0, T ) (see Section 2 for the explicit formulas), so that the evolution indeed corresponds to an evolution of closed curves with fixed total mass. The first boundary condition in (1.4) ensures that the rotation index is constant along the flow. The other boundary conditions are imposed in order to achieve that the corresponding curve is C 2 -closed and that the density is C 1 -periodic. Observe that the evolution equations are coupled so that we cannot treat them separately. An essential feature of working with the angle function θ instead of the curve γ is that the corresponding L 2 -gradient flow of the energy (1.2) is a parabolic system of second order in (θ, ρ). In contrast, the Euler-Lagrange equation of (1.1) as a functional of γ and ρ is of fourth order in γ, like the classical elastica equation. Such an approach using the angle function has been used for instance in [28] to prove well-posedness and global existence for a flow towards elastica. Local and global existence of solutions for the classical elastic flow, given by a parabolic fourth order equation in R n , has been shown in several works, for instance [13,20,11,9,19,29,8,27,21,25]. For a more detailed overview, see the recent survey [18]. A parabolic system related to (1.4) is studied in [1] and numerically elaborated in [12]. In the following, β ∈ C ∞ (R) satisfies β(x) > 0 for all x ∈ R and the model parameters L > 0, µ > 0, ω ∈ Z and c 0 ∈ R are fixed. We summarize the compatibility conditions for the initial datum. In the following existence result, h 1+α is the little Hölder space, cf. Appendix A.  Moreover, the solution depends continuously on the initial datum.
In the literature, short-time existence and uniqueness for quasilinear parabolic systems is widely accepted if the initial datum is smooth. Our contribution provides a complete proof of local well-posedness for the system (1.4), including continuous dependence on the data, even for initial data with low regularity and despite the presence of nonlocal terms, in a fairly concise way. The idea of the proof is to meet the boundary conditions by formulating the problem in a periodic setting. Then, following the ideas in [3], the Inverse Function Theorem between appropriate time-dependent little Hölder spaces yields existence and continuous dependence on the initial datum. We deduce the local invertibility of the Fréchet derivative from the maximal regularity theory developed in [16] and a perturbation argument. The precise formulation of the continuous dependence is given in Proposition 3.3 and Remark 3.4 below. To show instantaneous smoothing, we rely on the theory of parabolic systems in [14] and a bootstrap argument. Finally, exploiting the dissipative structure of the system, uniqueness follows from an energy argument. For the classical elastic flow, one expects long-time existence of the solution. We prove such a result in this much more general model. The flow equations (1.4) are a parabolic system, so we cannot use arguments based on the maximum principle to show global existence. Instead, we rely on energy estimates to prove that suitable Sobolev norms along the solution cannot blow up in finite time. To that end, we first show boundedness of the velocity ∂ t (θ, ρ) in order to deal with the nonlinear coupling which limits the direct application of interpolation inequalities. To allow for ω = 0, a new argument is needed to show a priori boundedness of the nonlocal Lagrange multipliers, compared to the strategy used in [28]. Once global existence is established, it is natural to study the asymptotic behavior of the flow. Although we do not have a classification of the stationary solutions to (1.4) at hand, we can prove full convergence. Theorem 1.4 (Long-time behavior). Let β : R → R be real analytic. Then, the global solution (θ, ρ) in Theorem 1.2 converges in C 2+α ([0, L]) for allα ∈ (0, 1 2 ) to a stationary solution of (1.2) as t → ∞, i.e. to a solution of for some λ θ1 , λ θ2 , λ ρ ∈ R.
From the energy estimates used in the proof of Theorem 1.3, we first deduce subconvergence of the solution. To prove full convergence, we assume β to be analytic, since we rely on a suitable version of the Lojasiewicz-Simon gradient inequality [22], a powerful functional analytic tool for studying the asymptotic behavior of (geometric) PDEs, even in the constrained setting; see, for instance, [26,6,10] and [25,23,24].
Geometrically, our results can be interpreted as follows. The solution (θ(t), ρ(t)) t≥0 describes a family of closed curves (γ(t)) t≥0 with density (ρ(t)) t≥0 that evolves in time decreasing the energy (1.1) and converging to an equilibrium as t → ∞. This article is structured as follows. In Section 2, we briefly review some basic geometric concepts concerning curves and give the formulas for the Lagrange multipliers. The goal of Section 3 is to prove Theorem 1.2. First, we prove a local well-posedness result in a periodic setting and then show that this periodic solution gives a solution to (1.4). Sections 4 and 5 are devoted to prove Theorem 1.3 and Theorem 1.4, respectively.

Preliminaries
We consider an arc-length parametrization of a closed regular curve γ : [0, L] → R 2 of length L > 0 in the plane, which we assume to be sufficiently smooth. By a standard result in elementary differential geometry, there exists a function θ : [0, L] → R such that ∂ s γ = (cos θ, sin θ). Such an inclination angle θ is not unique, in fact it is only unique up to addition of an integer multiple of 2π. Nevertheless, many geometric quantities of the curve can be expressed in terms of an inclination angle; the (signed) curvature κ of the curve γ is given by κ = ∂ s θ and its rotation index ω ∈ Z satisfies 2πω = θ(L) − θ(0). In (1.4) three Lagrange multipliers appear. In the first equation, λ θ1 and λ θ2 ensure that γ remains closed along the flow if the initial curve is closed. Indeed, defining where ψ(t, s,s) := θ(t, s) − θ(t,s). Hence, as long as θ(t, ·) is not constant, the matrix Π(θ)(t) is invertible. In particular, the Lagrange multipliers in (2.1) are well-defined, as long as the angle function describes a closed curve.
Remark 2.2. The boundary conditions in (1.4) make it possible to use integration by parts to write the Lagrange multipliers λ θ1 and λ θ2 in the form The third Lagrange multiplier λ ρ is chosen so that the total mass remains fixed during the flow. Defining a sufficiently smooth solution (θ, ρ) of (1.4) satisfies ∂ t L 0 ρ ds = 0. (2.5) From the fact that λ θ1 and λ θ2 ensure that the curve remains closed during the flow, we can already deduce that the integral over the inclination angle is preserved.
By (1.5), the integrals vanish. The boundary conditions in (1.4) yield the claim.
A direct computation reveals that we have an L 2 -gradient flow structure, i.e. for a sufficiently smooth solution (θ, ρ) of (1.4), we have

Local well-posedness and regularity
To prove Theorem 1.2, we will consider a periodic setting. This is quite natural due to the geometric interpretation of the problem. Doing so, we get rid of the boundary conditions which enables us to apply an existence result in [16] for a suitable linearization of our problem. We first prove short-time existence and continuous dependence on the data as well as uniqueness and smoothness of the solution to the periodic problem. Then, by a suitable transformation, we can transfer the results to problem (1.4).
3.1. Transformation to a periodic setting. For the moment, assume that (θ, ρ) is a sufficiently smooth function in (t, s). We observe that the boundary conditions in (1.4) imply that ∂ s θ is a C 0 -periodic function, i.e. it can be extended to a continuous L-periodic function on R. Similarly, ρ is a C 1 -periodic function. The angle function θ is itself not periodic if the rotation number is different from zero. We hence first perform a suitable transformation. Define the function  In terms of (u, ρ) L-periodic, (1.4) can be equivalently written as Here, to simplify the notation, instead of working with (λ θ1 , λ θ2 ) (see Remark 2.2) we consider Λ u given by for (t, s) ∈ (0, T ) × R and, instead of λ ρ , Λ ρ given by for t ∈ (0, T ), see (2.4). We now define our concept of a solution. To that end, we work with time-weighted little Hölder spaces (see Appendix A for a precise definition), which provide an appropriate framework for the local well-posedness of the problem.
The time weight (described by the parameter η) allows us to consider less regular initial data. For T > 0, α ∈ (0, 1) and η ∈ 1 2 , 1 such that 2η + α ∈ Z, we set for k = 1, 2. For k = 2, E 1 is the space of solutions in the sense of Definition 3.1. For (u, ρ) ∈ E 1 the right hand side of the evolution equations in (3.3) belongs to E 0 , cf. Remark A.5. We define γE 1 := h 2η+α L (R; R k ), the space for initial data (see (A.3)). In the following, the value of k = 1, 2 will be clear from the context. det Π(u + φ) > 0 and define the nonlinear differential operator The evolution equations in (3.3) can be written as ∂ t (u, ρ) = F(u, ρ) on (0, T ) × R.
Remark 3.2. The choice η > 1 2 ensures that the initial datum has finite energy, but also turns out to be essential to control the nonlinearities in the equation (cf. Lemma B.3, Proposition B.6) and to show uniqueness (cf. Lemmas 3.6, A.6).
3.2. Short-time existence and continuous dependence. To prove short-time existence and continuous dependence on the data, we essentially follow the ideas of [3,Theorem 2]. However, since we work with periodic little Hölder spaces, we have to justify some steps differently. Proposition 3.3 (Existence and continuous dependence). Let α ∈ (0, 1) and η ∈ 1 2 , 1 . Let (u 0 , ρ 0 ) ∈ γE 1 and let u 0 satisfy (3.2). Then there exist T 0 ∈ (0, ∞) and r > 0 such that for every admits a solution (ũ,ρ) ∈ E 1 ([0, T 0 ]). Moreover, there is a C 1 -map which associates to the initial datum a solution of the problem. In particular, there exists a solution Proof. We divide the proof into several steps. For the sake of readability, the details are partially moved to Appendix B. We will refer to this at the respective points.
Step 2: We claim that there exists T ′ ∈ (0,T ] such that the nonlinear operator is well-defined and a local diffeomorphism at (ū,ρ) (restricted to [0, T ′ ]). Here F is the map defined in (3.7). It is sufficient to prove the existence of T ′ ∈ (0,T ] such that the Fréchet derivative is a linear isomorphism. It then follows, since Φ ∈ C 1 , by the Local Inverse Function Theorem that Φ is a local diffeomorphism at (ū,ρ). The proof that the map in (3.11) is a linear isomorphism is given in Proposition B.6 (withJ = DΦ(ū,ρ)). The idea is to use a maximal regularity result in time-weighted little Hölder spaces [16], where the time weight allows for rough initial data. Since the results in [16] are applicable to linear systems with time-independent coefficients, we first consider an auxiliary linear problem, cf. Proposition B.1. A perturbation argument then gives that the map in (3.11) is an isomorphism for small times.
Step 3: By Step 2, there is a neighborhood V ⊂ U([0, T ′ ]) of (ū,ρ) and a neighbor- Here it is crucial that the E 0 -norm becomes arbitrary small for small times (see the definition of BUC 1−η in Section A.2).
Remark 3.4. In the following section we prove uniqueness of solutions. Together with Proposition 3.3 this yields the continuity -or more precisely even the continuous differentiability -of the mapping which maps any initial datum

Smoothness and uniqueness of solutions.
We now show that a solution is smooth for positive times.
The idea is to define a smooth cut-off function, which vanishes at t = 0 and outside of a spatial interval of length larger than L. The product of u with this cutoff function is the solution of a new initial value problem -this time with zero boundary values and zero initial data. This enables us to use [14, Theorem IV.5.2], which gives us higher regularity of u in the scale of parabolic Hölder spaces, cf. Appendix A.3. We proceed similarly for ρ and iterate this procedure to conclude smoothness.
We are now able to prove that solutions are unique.
Lemma 3.6 (Uniqueness). Let (u 0 , ρ 0 ) ∈ γE 1 . Let 0 < T 1 , T 2 ≤ ∞ and let (u 1 , ρ 1 ) and (u 2 , ρ 2 ) be solutions of (3.3) on (0, T 1 ) × R and (0, Proof. Without loss of generality we assume 0 is differentiable. In the following, C denotes a constant, only depending on u 1 , u 2 , ρ 1 , ρ 2 , T , and the model parameters β, c 0 , L, µ, ω, which may change from line to line. Other dependencies are given explicitly. Lemma After integration by parts and using We estimate everything but the first three terms using Young's inequality to absorb higher order derivatives and bound the remaining ones by ϕ(t) using (3.15). Indeed, since Hence, the term on the second line in (3.17) can be estimated by a constant times In the third line of (3.17), we add, subtract, and use Young's inequality to find On the other hand, for the term in the third line of (3.17) involving Λ u we have We write the term in square brackets as Using the explicit formula for the inverse matrix, it can be shown that Proceeding as in (3.19), after adding and subtracting, we have Consequently, (3.20) may be estimated by For the remaining terms, we proceed similarly. Using (3.16) and choosing δ > 0 sufficiently small, the first two terms in (3.17) can be used to absorb all terms involving a factor of δ. Altogether, we obtain the differential inequality Using Gronwall's lemma, we conclude that for all t ′ ∈ [0, T ] we have By Lemma A.6, the integral is finite and uniqueness follows from ϕ(0) = 0.
3.4. Proof of Theorem 1.2. It remains to transfer the results of Section 3.2 and Section 3.3 to the initial nonperiodic setting and to prove Theorem 1.2.
. This is an important ingredient in proving global existence, cf. Section 4.3 below.
4.2. Energy estimates. We now prove norm estimates for our solution, which by Lemma 3.5 instantaneously becomes smooth. Since we are only interested in long-time existence here, for the rest of this subsection we assume that we have a fixed smooth solution (θ, ρ) ∈ C ∞ ([0, T ) × [0, L]) of (1.4). This means that the initial datum (θ 0 , ρ 0 ) is smooth and satisfies Hypothesis 1.1. Furthermore, throughout this subsection C ∈ (0, ∞) denotes a generic constant that may vary from line to line. The constant is only allowed to depend on the fixed model parameters as well as the initial datum (θ 0 , ρ 0 ). In particular, C does not depend on the (not necessarily maximal) existence time T . This is essential for proving the convergence in Section 5.3. First, we gather some direct consequences of the constrained gradient flow structure.
Proof. Statement (iii) follows from integrating (2.6) in time and using E µ (θ, ρ) ≥ 0. Moreover, for any t ∈ [0, T ) we obtain and the second estimate in (ii) is proven.
We will now use this to get an L ∞ -bound on ρ. Since the total mass is conserved along the evolution by (1.  By continuity, we find that Therefore, using (2.6) again, we obtain

By (1.3), we have
L 0 ∂ s θ ds = 2πω, and hence the first part of (ii) follows. This together with the fact that where J is as in (4.3). The right hand side is bounded for all t ∈ [0, T ) by Proposition 4.2 and the Cauchy-Schwarz inequality. This way, we obtain the desired bounds for λ θ1 and λ θ2 . For λ ρ we use (2.4) and proceed similarly, using that sup J |β ′ | ≤ C by continuity.
As a next step, we would like to bound the L 2 -norm of ∂ 2 s (θ, ρ) uniformly in time. Directly pursuing this idea, we encounter difficulties in controlling the nonlinear coupling of the system (1.4), which seems to be incompatible with a direct application of interpolation inequalities. Instead, we control the L 2 -norm of the velocity ∂ t (θ, ρ) first.
Proof. By continuity, Proposition 4.2 and with J as in (4.3) we have that We consider the smooth function We now use (1.4) and integration by parts to differentiate ϕ. The boundary terms dissapear as a consequence of the boundary conditions in (1.4). We have for δ > 0 to be chosen and some correspondingly adjusted C(δ) > 0. Using (2.5) and arguing as in using Young's inequality in the last step. Using (4.4) and choosing δ > 0 small enough in (4.6) and (4.7), we obtain d dt ϕ(t) ≤ Cϕ(t) for all t ∈ [0, T ).
Proof. To prove the required estimates, we need to consider simultaneously the evolution equations of θ and ρ. Recall from (1.4) that we have . Using (4.4) and arguing as in (4.5), by Young's inequality we find Due to Proposition 4.2 the norm ∂ s θ L 2 (0,L) is bounded for all t ∈ [0, T ) and similarly for ρ. Therefore, using twice Young's inequality we estimate , for δ > 0 to be chosen. In an analogous way we also get (∂ s θ) 4 L 1 (0,L) ≤ δ ∂ 2 s θ 2 L 2 (0,L) + C(δ). Finally, taking δ > 0 sufficiently small and absorbing, the claim follows.

Convergence result
In this section, we prove the convergence result, Theorem 1.4. Our main ingredient is a constrained version of the Lojasiewicz-Simon inequality [22]. The constraint is given by the zero set of the functional G(θ, ρ) = for all t ≥ 0 with φ as in (3.1). To apply the results in [22], we need to work in Banach spaces which is why we consider the shifted energies E µ (u, ρ) = E µ (φ + u, ρ), G(u, ρ) = G(φ + u, ρ) for (u, ρ) ∈ W 2,2 per (0, L; R 2 ). Here we work only in the domain of the L 2 -gradient of the functionals, and not in the energy space W 1,2 (0, L). By a direct computation, the L 2 -gradients of E µ (with ∇E µ (u, ρ) = (∇ u E µ (u, ρ), ∇ ρ E µ (u, ρ))) and of the components of G are given by  The rest follows using that the composition and sum of analytic maps is again analytic and that any bounded multilinear map is analytic, see [22, Section 2.1].
In this notation, the evolution in (1.4) may abstractly be written as  7)) are precisely the constrained critical points of E subject to the constraint G = 0 by the method of Lagrange multipliers.

Appendix A. Function Spaces
Already in the introduction in Theorem 1.2 little Hölder functions appear. Later in Chapter 3 we work with time-weighted little Hölder spaces and in the proof of Lemma 3.5 additionally the parabolic Hölder spaces are used. Here we collect the definitions of these spaces and list some properties.
A.1. Little Hölder functions. For α ∈ R + \ Z, we denote by ⌊α⌋ the largest integer less than α and {α} := α − ⌊α⌋ ∈ (0, 1). Let k ∈ N. The space of L-periodic R k -valued functions over R which are continuous and ⌊α⌋-times continuously differentiable is denoted by C ⌊α⌋ L (R; R k ). The periodic Hölder space with exponent α is defined as and equipped with the norm f C α (R) := f C ⌊α⌋ (R) + f (⌊α⌋) α,R . Therewith, we define the periodic little Hölder space with exponent α over R by This is a closed subspace of C α L (R; R k ) and therefore a Banach space. In the following we collect some properties of periodic little Hölder functions. This type of functions plays a fundamental role in the classical maximal regularity theory, see for example [17].
L (R) and let g 2 ∈ h α2 L (R). Then Proof. This follows from (A.1) and a lengthy computation.
A.2. Time-weighted BUC spaces. Let E be an arbitrary Banach space. Let η ∈ (0, 1) and let T > 0. We define respectively. We use that the mapping is well defined, linear and continuous. This follows from the observation that for ξ ∈ E 1 ([0, T ]) and 0 < s < t and hence there exists lim t→0 ξ(t) in h α L (R), see [7,Remark 2.1]. We introduce the trace space γE 1 := Im(γ). In particular, γE 1 does not depend on T > 0. In [16,Section 5.1] it is shown that up to equivalent norms provided 2η + α ∈ Z. For simplicity, we will mostly omit the index k. If η > 1 2 , we have continuity in space and time of the first order s-derivative by the following lemma. This is essential in Section 3. In particular, this implies that for We observe that from the definition of the spaces E 0 ([0, T ]) and E 1 ([0, T ]) we can derive the following properties. ( 2 ) and the finiteness of the norm Appendix B. Auxiliary results for Section 3.2 In the following, we show Step 2 of the proof of Proposition 3.3, i.e. that DΦ(ū,ρ) (see (3.11)) is a linear isomorphism. This is equivalent to show existence and uniqueness of a solution for a certain inhomogeneous linear initial value problem.
In this section, α ∈ (0, 1) and η ∈ 1 2 , 1 are fixed. B.1. The linearization. We compute here DF(ū,ρ) needed in (3.11), where F is given as in (3.7). By a direct calculation one finds Let T > 0. We first observe that by (3.4), in the Lagrange multipliers only spatial derivatives of first order appear. More specifically, Λ u is well defined with Λ u (u, ρ) ∈ 3), any initial datum (v 0 , σ 0 ) ∈ γE 1 is in F([0, T ]) if we identify it with the function (t, s) → (v 0 (s), σ 0 (s)). It is not difficult to see that is of class C 1 . Moreover, it follows from the product rule that B.2. Linear evolution problem with time-independent coefficients. We consider an auxiliary linear evolution problem. Since we rely on [16], where timeindependent coefficients are used, we would like to freeze the coefficients of the terms in (B.1) and (B.2) in t = 0. For the second term in (B.1), the regularity of the initial datum does not allow this. Therefore, we treat this term differently from the others and move it together with DΛ u and DΛ ρ , which are nonlocal, to the right hand side of the inhomogeneous initial value problem we want to solve. This explains the structure of the differential operator chosen below.
From the fact that the mapping J from Proposition B.1 is an isomorphism, an estimate of the norm of the solution follows implicitly. We now show that the constant in this estimate can be chosen independent of T .
be the map constructed in the proof of Proposition 3.3. Let T ∈ (0,T ]. Then the solution operator J −1 satisfies the estimate Notice that here we do not write explicitely the dependence on T of the solution operator. This abuse of notation is justified in the proof below. 1 Here we slightly abuse notation, denoting by (v 0 , σ 0 ) both the initial datum of the problem (B.6) and the evaluation at t = 0 of (v, σ) ∈ E 1 ([0, T ]). However, for solving (B.6), this does not make a difference.
Lemma B.5. Let M > 0 be as in (B.7). Then there exists a constant C, only depending on M , η, α and the model parameters, such that for all T ∈ (0,T ], Proof of Lemma B.5. In the following, C denotes a constant which varies from line to line but only depends on M and the model parameters. First, we prove (B.9) considering each term in (B.5) separately. By (2.2) with θ =ū + φ, we find DΠ (ū + φ) (v) = L 0 sin (2 (ū + φ)) v ds − L 0 cos (2 (ū + φ)) v ds − L 0 cos (2 (ū + φ)) v ds − L 0 sin (2 (ū + φ)) v ds (B.10) and obtain using (B.7) and (B.10) (considering the operator norm of the matrix) Hence, with Lemma B.3 the difference coming from the first term on the right hand side of (B.5) can be estimated by For the second term of (B.5) we integrate once by parts and obtain DJ (ū,ρ)(v, σ) = L 0 − sin (ū + φ) cos (ū + φ) v (∂ sū + ∂ s φ) β(ρ) (∂ sū + ∂ s φ − c 0 ) ds So the difference we need to consider can be estimated by In the last step we used Lemma B.3. Using (B.7), we obtain Finally, we look at the difference coming from the third term of (B.5). We have Thereby we obtain using (B.7) and Lemma B.3 This proves the first part of the claim. The second part follows similarly.
B.4. Solving the linearized problem. We show that the map in (3.11) is an isomorphism, now considering a linear initial value problem with time-dependent coefficients.
Proposition B.6. Let (u 0 , ρ 0 ) ∈ γE 1 such that u 0 satisfies (3.2) and let (ū,ρ) ∈ E 1 ([0,T ]) be the function constructed in the proof of Proposition 3.3. Then there exists T ′ ∈ (0,T ] such that the mapping is an isomorphism. Equivalently, for any right-hand side (ϕ 1 , ϕ 2 ) ∈ E 0 ([0, T ′ ]) and any initial datum (v 0 , σ 0 ) ∈ γE 1 there is a unique solution (v, σ) ∈ E 1 ([0, Proof. Let T ∈ (0,T ] to be chosen. We compareJ with J from Proposition B.1, which we already know is an isomorphism. We havẽ Thus, by a Neumann series argument, it is sufficient to show that the operator norm of S becomes arbitrarily small for T getting small and at the same time the operator norm J −1 remains bounded, independent of T ≤T . This last part has already been proven in Lemma B.2. Hence, the statement follows if we show S ≤ C max T, T η , Tη . (B.11) By definition of A, Ψ and using (B.1), (B.2) we find We now take a look at the terms of (B.12) and (B.13). In the following, C denotes a constant, only depending on the model parameters, α, η and M as in (B.7), which is allowed to change from line to line. The E 0 -norm of the first term of (B.12) can be estimated using Lemma A.2 and Lemma B.4 by Here we used the time-weight of the E 0 -norm to bound the second derivative of v.
For the second term of (B.12) we have to use the time-weight to bound the second derivative ofū. An additional factor depending on T and going to zero for T going to zero is given by Lemma B.3. This motivates the choice of the first term in Ψ in Proposition B.1. We obtain The third term of (B.12) can be estimated using Lemma B.4 and Lemma B.3 by The same also applies for the fourth term of (B.12) and the second term of (B.13). For the fifth term of (B.12) we proceed similarly and estimate Analogously, we can treat the first term of (B.13). Together with Lemma B.5, (B.11) and thus the claim follows for T ′ = T <T sufficiently small.