Abstract
We show that solutions for a specifically scaled nonlinear wave equation of nonlinear elasticity converge to solutions of a linear Euler–Bernoulli beam system. We construct an approximation of the solution, using a suitable asymptotic expansion ansatz based upon solutions to the one-dimensional beam equation. Following this, we derive the existence of appropriately scaled initial data and can bound the difference between the analytical solution and the approximating sequence.
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1 Introduction
The relation between solutions of the equations of nonlinear elasticity and solutions for lower-dimensional models is of great interest since the lower-dimensional models are often easier to analyze and to use for numerical simulations. A general introduction to this topic can be found in [5] or for continuum mechanics see [9]. In dependence of the size of the deformation and the applied forces, different lower-dimensional models can occur. Therefore, a rigorous derivation of the lower-dimensional models is of great interest. In the case of the time-independent case, there are many results, cf. e.g., Friesecke et al. [7, 8] for the case of plates and Mora and Müller [11, 12] in the case of rods and Scardia [14, 15] for curved rods. But there are only few results in the time-independent case so far.
In this contribution we investigate the relation between solutions of an appropriately scaled wave equation of nonlinear elasticity and solutions of a linear Euler–Bernoulli beam system. More precisely, let \(\Omega := [0,L]\times S\) be the reference configuration of a three-dimensional rod, where \(L>0\) and \(S\subset \mathbb {R}^2\) is the cross section. Then we consider the following nonlinear system
where \(T > 0\) and \(\tilde{W}\) is some elastic energy density chosen later. Existence of strong solutions for large times of this system was shown in [1]. More details on how to justify the scaling can be found there as well. The limit system as \(h\rightarrow 0\) is given by
where \(\tilde{v}_0, \tilde{v}_1\) and \(I_2, I_3\) are appropriately chosen initial values and weights, respectively. It is the goal of this manuscript to prove convergence of the solutions of the system (1.1)–(1.4) to solutions of the limit system (1.5)–(1.7) with appropriate convergence rates for well-prepared initial data, cf. Theorem 3.3 below.
In an energetic setting the relations between higher-dimensional models and lower-dimensional ones, using the notion of \(\Gamma \)-convergence a fundamental contribution was given in [7]. There the classical geometric rigidity is proven. Using this result it was possible to prove a lot of convergence results in different geometrical situations and scaling regimes in the static setting, see, for instance, [8, 11, 12]. In the dynamical case for plates, a convergence result can be found in [3] and in [6] in the case of viscoelasticity. The large times existence and a first-order asymptotic for plates were shown in [2].
In the following we want to explain the main novelties and difficulties of this contribution. In a first step we construct an approximation using the solution of the lower-dimensional system. This approximation is constructed such that it solves the linearization around zero of the nonlinear, three-dimensional equation up to an error of order \(h^3\). This is done explicitly by determining suitable prefactor functions as solutions of systems on S. Thereafter, the main difficulty of this work is to establish existence of suitable initial data in order to ensure large times existence for the solution of the nonlinear problem. This is done in Sect. 3.2. Here we use the nonlinear equations for the initial data from the compatibility conditions. These are solved via a fixed point argument on precisely chosen function spaces. Finally, in Sect. 3.3 the convergence properties are proven. For this we use a general result for solutions of the linearized equation. Moreover, we have to carefully treat the rotational parts of the initial data, as the spaces for the fixed point argument do not cover them. For this we use a decomposition and the fact that the elastic energy density is chosen as \(\tilde{W}(F) = {\text {dist}}(Id+F; SO(3))\).
The results are part of the second author’s PhD thesis [4].
2 Preliminaries and auxiliary results
2.1 Notation
We use standard notation; in particular \(\mathbb {N}\) and \(\mathbb {N}_0 := \mathbb {N}\cup \{0\}\) denote the natural numbers without and with zero, respectively. Moreover, the norm on \(\mathbb {R}\) and absolute value in \(\mathbb {R}^n\), \(\mathbb {R}^{n\times n}\) is denoted by |.| for all \(n\in \mathbb {N}\). For \(p,~k\in \mathbb {N}\), we denote the classical Lebesgue and Sobolev spaces for some bounded, open set \(M\subset \mathbb {R}^n\), by \(L^p(M)\), \(W^k_p(M)\) and \(H^k(M) := W^k_2(M)\). A subscript (0) on a function space will always indicate that elements have zero mean value, e.g., for \(g\in H^1_{(0)}(M)\) we have
The cross section of the rod is always denoted by \(S\subset \mathbb {R}^2\) and is assumed to be a smooth and bounded domain. Furthermore, let \(\Omega _h := (0,L) \times hS \subset \mathbb {R}^3\) for \(h \in (0,1]\) and \(L>0\) and for convenience we write \(\Omega := \Omega _1\). We assume that S satisfies
where \(x' := (x_2, x_3)\subset \mathbb {R}^2\). This is no loss of generality, as it can always be achieved via a translation and rotation. The scaling shell is such that we can assume \(|S|=1\). Furthermore, we denote with \(\nabla _h\) the scaled gradient defined as
The respective gradient in only \(x':=(x_2, x_3)\) direction is denoted by
The standard notation \(H^k(\Omega )\) and \(H^k(\Omega ; X)\) is used for \(L^2\)-Sobolev spaces of order \(k\in \mathbb {N}\) with values in \(\mathbb {R}\) and some space X, respectively.
The space of all n-linear mappings \(G:V^n \rightarrow \mathbb {R}\) for a vector space V is denoted, throughout the paper by \(\mathcal {L}^n(V)\), \(n\in \mathbb {N}\). We deploy the standard identification of \(\mathcal {L}^1(\mathbb {R}^{n\times n}) = (\mathbb {R}^{n\times n})'\) with \(\mathbb {R}^{n\times n}\), i.e., \(G\in \mathcal {L}^1(\mathbb {R}^{n\times n})\) is identified with \(A\in \mathbb {R}^{n\times n}\) such that
where \(A:X = \sum _{i,j=1}^n a_{ij} x_{ij}\) is the usual inner product on \(\mathbb {R}^{n\times n}\). Analogously, for \(G\in \mathcal {L}^2(\mathbb {R}^{n\times n})\) we use the identification with \(\tilde{G}:\mathbb {R}^{n\times n}\rightarrow \mathbb {R}^{n\times n}\) defined by
We introduce a scaled inner product on \(\mathbb {R}^{n\times n}\)
for all A, \(B\in \mathbb {R}^{n\times n}\) and \(h>0\) and the corresponding norm is denoted by \(|A|_h := \sqrt{A:_hA}\). With this we can define for \(W\in \mathcal {L}^d(\mathbb {R}^{n\times n})\) the induced scaled norm by
Using \(|A|_h \ge |A|_1 = |A|\) for all \(A\in \mathbb {R}^{n\times n}\), it follows \(|W|_h \le |W|_1 =: |W|\) for all \(W\in \mathcal {L}^d(\mathbb {R}^{n\times n})\) and \(0 < h \le 1\). The scaled \(L^p\)-spaces are defined as follows
if \(p\in [1, \infty )\), where \(U\subset \mathbb {R}^d\) is measurable. Thus \(\Vert W\Vert _{L^p_h(U; \mathcal {L}^d(\mathbb {R}^{n\times n}))} \le \Vert W\Vert _{L^p(U;\mathcal {L}^d(\mathbb {R}^{n\times n}))}\). The scaled norm for \(f\in L^p(U, \mathbb {R}^{n\times n})\) is defined in the same way
Then
As we will work with periodic boundary condition in \(x_1\)-direction, we introduce for \(m\in \mathbb {N}\)
This space can equivalently defined in the following way, which is in some situations more convenient
We equipped \(\tilde{H}^m_{\textrm{per}}(\Omega )\) with the standard \(H^m(\Omega )\)-norm. As the maps \(f\mapsto f|_{\Omega }:\tilde{H}^m_{\textrm{per}}(\Omega ) \rightarrow H^m_{\textrm{per}}(\Omega )\) and \(f\mapsto f_{\textrm{per}}:H^m_{\textrm{per}}(\Omega ) \rightarrow \tilde{H}^m_{\textrm{per}}(\Omega )\) are isomorphisms, we identify \(\tilde{H}^m_{\textrm{per}}(\Omega )\) with \(H^m_{\textrm{per}}(\Omega )\). This leads immediately to the density of smooth functions in \(H^m_{\textrm{per}}(\Omega )\), because, as S is smooth, there exists an appropriate extension operator and thus we can use a convolution argument.
In various estimates we will use an anisotropic variant of \(H^k(\Omega )\), as we will have more regularity in lateral direction. Therefore, we define
where \(m_1,~m_2\in \mathbb {N}_0\), the inner product is given by
Furthermore, we will use the scaled norms
for \(A\in H^m(\Omega ;\mathbb {R}^{n\times n})\) and \(B\in H^{m_1,m_2}(\Omega ;\mathbb {R}^{n\times n})\) and \(n\in \mathbb {N}\). As an abbreviation we denote for \(u\in H^k(\Omega ;\mathbb {R}^3)\) the symmetric scaled gradient by \(\varepsilon _h(u):={\text {sym}}(\nabla _h u)\) and \(\varepsilon (u) = \varepsilon _1(u) = {\text {sym}}(\nabla u)\).
The following lemma provides the possibility to take traces for \(u\in H^{0,1}(\Omega )\):
Lemma 2.1
The operator \({\text {tr}}_{a}:H^{0,1}(\Omega ) \rightarrow L^2(S)\), \(u\mapsto u|_{x_1=a}\) is well defined and bounded.
Proof
This is an immediate consequence of the embedding
where BUC([0, L]; X) is the space of all uniformly continuous functions \(f:[0,L] \rightarrow X\) for some Banach space X. \(\square \)
2.2 The strain energy density W and Korn’s inequality
We investigate the mathematical assumptions and resulting properties of the strain-energy density W we use in this contribution. We assume to have \(W:\mathbb {R}^{3\times 3}\rightarrow [0,\infty )\) defined by
where SO(3) denotes the group of special orthogonal matrices. This energy density clearly satisfies the following general assumptions
-
(i)
\(W\in C^\infty (B_\delta (Id); [0,\infty ))\) for some \(\delta >0\);
-
(ii)
W is frame-invariant, i.e., \(W(RF) = W(F)\) for all \(F\in \mathbb {R}^{3\times 3}\) and \(R\in SO(3)\);
-
(iii)
there exists \(c_0 > 0\) such that \(W(F) \ge c_0 {\text {dist}}(F, SO(3))^2\) for all \(F\in \mathbb {R}^{3\times 3}\) and \(W(R) = 0\) for every \(R\in SO(3)\).
Remark 2.2
We note that W has a minimum point at the identity, as \(W(Id) = 0\) and \(W(F)\ge 0\) for all \(F\in \mathbb {R}^{3\times 3}\). Hence, we have for \(\tilde{W}(F):=W(Id + F)\) for all \(F\in \mathbb {R}^{3\times 3}\), \(D\tilde{W}(0)[G] = 0\) for all \(G\in \mathbb {R}^{3\times 3}\). Moreover, it holds \(D^2 \tilde{W}(0) F = {\text {sym}}F\) and for \(P\in \mathbb {R}^{3\times 3}_{skew}\), \(A, B\in \mathbb {R}^{3\times 3}\) we obtain
The following lemma provides an essential decomposition of \(D^3\tilde{W}\) in the general form.
Lemma 2.3
There is some constant \(C>0\), \(\varepsilon >0\) and \(A\in C^\infty (\overline{B_\varepsilon (0)}; \mathcal {L}^3(\mathbb {R}^{n\times n}))\) such that for all \(G\in \mathbb {R}^{n\times n}\) with \(|G| \le \varepsilon \) we have
where
Proof
For the proof we refer to [2, Lemma 2.6]. \(\square \)
With this we can prove the following bound for \(D^3\tilde{W}\).
Corollary 2.4
There exist C, \(\varepsilon > 0\) such that
for all \(Y_1\in H^2(\Omega , \mathbb {R}^{n\times n})\), \(Y_2\), \(Y_3\in L^2(\Omega ; \mathbb {R}^{n\times n})\), \(0<h\le 1\) and \(\Vert Z\Vert _{L^\infty (\Omega } \le \min \{\varepsilon , h\}\) and
for all \(Y_1\), \(Y_2\in H^1(\Omega , \mathbb {R}^{n\times n})\), \(Y_3\in L^2(\Omega ; \mathbb {R}^{n\times n})\), \(0<h\le 1\) and \(\Vert Z\Vert _{L^\infty (\Omega } \le \min \{\varepsilon , h\}\) and
for all \(Y_1\in L^\infty (\Omega , \mathbb {R}^{n\times n})\), \(Y_2\), \(Y_3\in L^2(\Omega ; \mathbb {R}^{n\times n})\), \(0<h\le 1\) and \(\Vert Z\Vert _{L^\infty (\Omega } \le \min \{\varepsilon , h\}\).
Proof
The inequalities follow directly from Lemma 2.3 and Hölder’s inequality. \(\square \)
In order to bound the full scaled gradient \(\nabla _h g\) of some function \(g\in H^1_{\textrm{per}}(\Omega )\) by the symmetric one, we need a sharp Korn’s inequality for thin rods. As rigid motions \(x\mapsto \alpha x^\perp \) for \(\alpha \in \mathbb {R}\) arbitrary are admissible functions in \(H^1_{\textrm{per}}(\Omega )\), we cannot expect that the full scaled gradient is bounded by \(\varepsilon _h(g)\). Precisely, we obtain the following results.
Lemma 2.5
There exists a constant \(C=C(\Omega )>0\) such that for all \(0 < h\le 1\) and \(u\in H^1_{\textrm{per}}(\Omega ;\mathbb {R}^3)\) we have
where
with \(a(u) = \frac{1}{|\Omega |}\int \limits _\Omega \partial _{x_3} u_2(x) - \partial _{x_2} u_3(x) \textrm{d}x\).
Proof
The proof is similar to [2, Lemma 2.1] and is done in [4, Lemma 2.4.4] \(\square \)
Lemma 2.6
(Korn inequality in integral form) For all \(0 < h\le 1\) and \(u\in H^1_{\textrm{per}}(\Omega ;\mathbb {R}^3)\), there exists a constant \(C_K=C_K(\Omega )\), such that
where \(x^\perp = (0, -x_3, x_2)^T\).
Proof
A proof can be found in [1]. \(\square \)
3 First-order expansion in a Linearized regime
We construct an approximation to the unique solution of the nonlinear system
where \(\tilde{W}(F) = W(Id+F)\) for all \(F\in \mathbb {R}^{3\times 3}\), \(T > 0\). We assume that
for some \(g\in \bigcap _{k=0}^3 W^k_1(0,T; H^{10-2k}_{\textrm{per}}(0,L; \mathbb {R}^2))\), which implies
for \(k=2, 3\). Moreover, we assume that
where \(M>0\) is chosen later. Without loss of generality we can assume \(\int \limits _0^L g dx_1 = 0\). Otherwise, we subtract
from \(u_h\) analogously as in the proof of [1, Theorem 3.1].
3.1 Construction of the ansatz function
For the ansatz function we consider the following system of one-dimensional beam equations
where \(\tilde{v}_{0}\in H^{12}_{\textrm{per}}(0,L;\mathbb {R}^2)\), \(\tilde{v}_{1}\in H^{10}_{\textrm{per}}(0,L;\mathbb {R}^2)\) such that
and
Then we obtain with standard methods, as, e.g., in [13, Theorem 11.8], the existence of a unique solution
Moreover, due to the assumptions for g and the periodicity of v it follows
Now we define
where a, b, \(c:S\rightarrow \mathbb {R}^2\) are chosen later. Then
Thus, with \(D^2 W(Id) F = {\text {sym}} F\) we can derive
for
Moreover, for the boundary condition it holds
We choose now \(a:S\rightarrow \mathbb {R}^2\) as the solution of the following system
with
Such a solution exists, because we can apply the Lax–Milgram Lemma for the weak Laplacian on \(H^1_{(0)}(S;\mathbb {R}^2)\). Thereby, the coercivity follows from Poincaré’s inequality. With well-known regularity result, e.g., Theorem 4.18 in [10], we obtain \(a\in C^\infty ({\overline{S}},\mathbb {R}^2)\). The systems for b and c decouple to
and
Defining the matrix of coefficients \(({\mathfrak {p}}^{\alpha \beta }_{ij})^{\alpha , \beta = 2,3}_{i,j=1,2}\) in the following way
With \(w = (b_2, c_2)^T\) and \(f = (-I_1 - \partial _{x_2} a_2, -\partial _{x_3} a_2)^T\), (3.8) is equivalent to
for \(i=1\), 2. Let now
be arbitrary. Then it holds
and thus \({\mathfrak {p}}^{\alpha \beta }_{ij}\) satisfies the Legendre condition for \(\lambda = \frac{1}{4}\). Thus, we can solve (3.8) and (3.9) with homogeneous Dirichlet boundary condition
as the system (3.9) can be treated in the same manner. The regularity of a implies now that \(b = (b_2,b_3)\) and \(c=(c_2,c_3)\) are \(C^\infty ({\overline{S}};\mathbb {R}^2)\).
The approximating solution \(\tilde{u}_h\) solves then the following system
where \(r_h\) is chosen as above,
and the initial data is given by
with \(v^j := \partial _t^j v|_{t=0}\) and \(j=0,\ldots , 4\). For the remainder it holds
3.2 Existence of and bounds on initial values
Define now
equipped with the norm
Lemma 3.1
There exist constants \(C_0>0\) and \(M_0\in (0,1]\) such that for \(0<h\le 1\) and \(f\in H^{1,1}_{\textrm{per}}(\Omega ;\mathbb {R}^3)\) with \(\Vert f\Vert _{H^{1,1}(\Omega )} \le M_0 h\) and \(\int \limits _\Omega f dx = 0\) there exists a unique solution \(w\in H^3_{\textrm{per}}(\Omega ; \mathbb {R}^3)\cap \mathcal {B}\) with \(\partial _{x_1} w\in H^3_{\textrm{per}}(\Omega ;\mathbb {R}^3)\) of
Moreover
holds. If \(w'\in H^3_{\textrm{per}}(\Omega ;\mathbb {R}^3)\cap \mathcal {B}\) with \(\partial _{x_1} w\in H^3_{\textrm{per}}(\Omega ;\mathbb {R}^3)\) is the solution to \(f'\in H^{1,1}_{\textrm{per}}(\Omega ;\mathbb {R}^3)\) with \(\Vert f'\Vert _{H^{1,1}(\Omega )} \le M_0 h\) and \(\int \limits _\Omega f' dx = 0\), then it holds
Proof
Using a Taylor series expansion for \(D\tilde{W}(\nabla _h w)\), we obtain
Thus, (3.11) is equivalent to
The idea is now to use the contraction mapping principle in order to prove the existence of a solution for (3.11), i.e., with the later equivalence
holds with \(G_h(w) := \frac{1}{h^2} G(\nabla _h w)\). Consequently, we investigate the mapping properties of \(L_h\) and \(G_h\).
For \(f\in L^2(\Omega ;\mathbb {R}^3)\) and \(F\in L^2(\Omega ;\mathbb {R}^{3\times 3})\), we obtain with the Lemma of Lax–Milgram the existence of a unique solution \(w\in \mathcal {B}\) for
for all \(\varphi \in \mathcal {B}\). The solution satisfies
If now \(f\in H^{0,k}(\Omega ;\mathbb {R}^3)\) and \(F\in H^{0,k}(\Omega ;\mathbb {R}^{3\times 3})\) for \(k=1,2\), it follows by a different quotient argument that \(w\in H^{0,k}(\Omega ;\mathbb {R}^3)\) holds and
Using the decomposition \(\mathcal {B}\oplus {\text {span}}\{x\mapsto x^\perp \} = H^1_{(0),per}(\Omega ;\mathbb {R}^3)\), it follows that for
we have
for all \(\varphi \in H^1_{(0),\textrm{per}}(\Omega ;\mathbb {R}^3)\). Hence, if \(f\in H^1_{\textrm{per}}(\Omega ;\mathbb {R}^3)\) and \(F\in H^2_{\textrm{per}}(\Omega ;\mathbb {R}^{3\times 3})\), then w solves the system
in a weak sense. Thus, with elliptic regularity theory it follows \(w\in H^3_{\textrm{per}}(\Omega ;\mathbb {R}^3)\cap \mathcal {B}\). By Theorem A.3 in the appendix, we obtain
where we have exploited
Using that \({\text {tr}}_{\partial S}:H^2(S) \rightarrow H^{\frac{3}{2}}(\partial S)\) is a bounded operator, we obtain
because of
Thus, we deduce for some \(C_L > 0\)
We define \(\mathcal {X}_h := H^3_{\textrm{per}}(\Omega ;\mathbb {R}^3)\cap \mathcal {B}\) and \(\mathcal {Y}_h := H^1_{\textrm{per}}(\Omega ;\mathbb {R}^3) \times H^2_{\textrm{per}}(\Omega ;\mathbb {R}^{3\times 3})\) normed via
This \(L_h^{-1}:\mathcal {Y}_h \rightarrow \mathcal {X}_h\) is a bilinear, bijective and bounded operator, mapping a tuple \((f,F)\in \mathcal {Y}_h\) to the corresponding solution \(w\in \mathcal {X}_h\) of (3.15). In order to close the proof, we have to show that \(G_h\) is a contraction with respect to the relevant norms.
In a first step we assume that \(w_i\in \mathcal {X}_h\) with
for \(i=1,2\) and \(M_1 > 0\) to be chosen later. Then
where we used Corollary 2.4, \(\Vert \nabla _h w_j\Vert _{H^1_h(\Omega )}\le C\Vert w_j\Vert _{\mathcal {X}_h}\) and the boundedness of
The definition of G implies that for \(k=1,2,3\) it holds
Hence, analogously as above
as
for \(\varphi = w_1\) and \(\varphi = w_1 - w_2\). With the aid of (3.18) it follows for \(j, k=1, 2, 3\)
Thus, we obtain in the same manner as above
The fact that \(h^2\Vert \nabla _h F\Vert _{H^1(\Omega )} \le h \Vert \nabla F\Vert _{H^1(\Omega )}\) and \(\Vert F\Vert _{L^2(\Omega )} \le \frac{1}{h}\Vert F\Vert _{(L^2_h)'}\) implies with the later estimates that for \(M_1\in (0,1]\) small enough
is a \(\frac{1}{2}\)-contraction. The self-mapping property of \(\mathcal {G}_{h,f}\) follows because of
Thus, we can choose \(M_0 > 0\) so small that \(C_L M_0 h \le \frac{C M_1}{2}\). Then we obtain with the \(\frac{1}{2}\)-contraction property of \(\mathcal {G}_{h,f}\) for \(w\in \overline{B_{CM_1 h}(0)}\)
Therefore, (3.12) and (3.13) hold with the \(H^{1,1}(\Omega )\)-norm on the left-hand side replaced by the \(\mathcal {X}_h\)-norm.
Using the decomposition \(\mathcal {B}\oplus {\text {span}}\{x\mapsto x^\perp \} = H^1_{(0),per}(\Omega ;\mathbb {R}^3)\), it follows that for
we have
for all \(\varphi \in H^1_{(0),per}(\Omega ;\mathbb {R}^3)\). If now \(f\in H^{1,1}_{\textrm{per}}(\Omega ;\mathbb {R}^3)\) we obtain, with a difference quotient argument, that \(w\in H^3_{\textrm{per}}(\Omega ;\mathbb {R}^3)\cap \mathcal {B}\) satisfies
for all \(\varphi \in H^1_{(0),per}(\Omega ;\mathbb {R}^3)\). Thus, with Theorem A.3 the claimed inequalities follow. \(\square \)
We define the initial values for the analytical problem as
for \(j=1,2\) and \(v^{2+j} = \partial _t^{2+j} v|_{t=0}\) as above.
Lemma 3.2
Let \(\tilde{u}_h\) be as in (3.7), \(\tilde{u}_{j,h}\) for \(j=0,1,2\) as in (3.10), \(u_{3,h}\), \(u_{4,h}\) and \(f^h\) be as above. Then for sufficiently small \(h_0\in (0,1]\) and \(M>0\), there exist solutions \((u_{0,h}, u_{1,h}, u_{2,h})\) of
and
for all \(\varphi \in \mathcal {B}\), where
and
The solution satisfies
and \(u_{k,h}\in \mathcal {B}\) for \(k=0,1,2\). Moreover, we have
for all \(h\in (0,h_0]\) and \(C>0\) independent of h.
Proof
We can equivalently formulate (3.19)–(3.21) via
and
for all \(\varphi \in \mathcal {B}\), where \(u_{0,h} = \mathcal {G}_{h,f}(u_{0,h})\) is the solution of (3.19) with \(f = h^2 f^h - u_{2,h}\). Defining
and deploying (3.13) we obtain for \(u_{2,h}\), \(u'_{2,h} \in H^{1,1}(\Omega ;\mathbb {R}^3)\)
if \(\Vert u_{2,h}\Vert _{H^{1,1}(\Omega )} \le \frac{1}{2}M_0 h\), \(\Vert u'_{2,h}\Vert _{H^{1,1}(\Omega )} \le \frac{1}{2}M_0 h\) and \(h^2\Vert f^h\Vert _{H^{1,1}(\Omega )}\le \frac{1}{2}M_0 h\). This can always be achieved if \(h_0\in (0,1]\) is small enough and \(u_{2,h}\), \(u'_{2,h}\) are of order \(h^2\).
Using the definition of \(L_h\) it follows that (3.25)–(3.26) are equivalent to
and
for all \(\varphi \in \mathcal {B}\). Defining now the relevant function spaces by
with the respective norms defined by
With this we define the linear operator \(\mathcal {L}_h^{-1}:\mathcal {Z}_h \rightarrow \mathcal {W}_h\) by mapping \((f_1,f_2, F_1, F_2)\) to the solution \((w_1, w_2)\) of
for \(i=1,2\). Then due to (3.17), Theorem A.3 and (3.16) we obtain
Hence, \(\mathcal {L}_h^{-1}\) is a bijective, linear and bounded operator. For the nonlinearity we define
via
where \(u_0 := \mathcal {G}_{h,f - u_2}(u_{0,h})\) for some fixed \(f\in H^{1,1}_{\textrm{per}}(\Omega )\) with \(\Vert f\Vert _{H^{1,1}(\Omega )} \le M h^2\) and \(\int \limits _\Omega f dx = 0\) and G is defined as in (3.14).
We deduce the contraction properties of \(\mathcal {Q}_h\) similar as in the proof of Lemma 3.1. For this we assume that \(\Vert (u_1, u_2)\Vert _{\mathcal {W}_h}\), \(\Vert (u'_1, u'_2)\Vert _{\mathcal {W}_h}\le CM_2 h^2\). Starting with \(\mathcal {Q}_{1,h}\), we obtain
where we used (3.27). Similarly one deduces that
for \(j,k=1,2,3\). Analogously we deduce for \(\mathcal {Q}_{2,h}\)
where we used again Corollary 2.4, \(|P|_h = |P|\), \(|\gamma _h(u_{0,h})| \le Ch^2\) and
Finally from
it follows
Choosing now \(M_2 \in (0,1]\) small enough we obtain that
defined by
is a \(\frac{1}{2}\)-contraction, where \(f_0 := h^2 f^h|_{t=0}\), \(f_1 := h^2\partial _t f^h|_{t=0} - u_{3,h}\) and \(f_2 := h^2\partial ^2_t f^h|_{t=0} - u_{4,h}\). We can use an analogous argument as in Lemma 3.1. First it holds, due to (3.6) and (3.5), for \(M>0\) sufficiently small
and with the \(\frac{1}{2}\)-contraction property we obtain the self mapping of \(\mathcal {F}_{h, f_0,f_1,f_2}\). Moreover, due to the norm on \(\mathcal {X}_h\) and \(\mathcal {W}_h\) we obtain (3.22) and (3.23), respectively.
Finally, the construction of \(\tilde{u}_h\) implies that \(\tilde{u}_{j,h}\) satisfies
for \(j=0,1,2\) and all \(\varphi \in \mathcal {B}\). This implies with (3.19)–(3.21)
for all \(\varphi \in \mathcal {B}\), where we defined
With this it follows \(\max _{j=1,2} \Vert r_{j,h}\Vert _{C^0(0,T;L^2(\Omega ))}\le Ch^3\) because of the definition of \(u_{2+j,h}\) and the bound on \(\partial _t r_h\). Additionally, we have due to Lemma 2.3 and Corollary 2.4, the bounds on \((u_{0,h}, u_{1,h}, u_{2,h})\) and \(\varphi \in \mathcal {B}\)
as well as
and
Regarding the boundary terms, we use that \({\text {tr}}_{\partial S}:H^1(S)\rightarrow H^{\frac{1}{2}}(\partial S)\) is linear and bounded. Hence, for \(j=0,1,2\)
where we used that \(\Vert r_{N,h}\Vert _{C^2([0,T]; H^1(\Omega ))} \le Ch^5\) and the Poincaré and Korn inequality for \(\varphi \). Choosing \(\varphi = u_{j,h} - \tilde{u}_{j,h}\), it follows with an absorption argument
Now, for \(u_{0,h}-\tilde{u}_{0,h}\) it holds
The definition of G implies now
because of the bounds for \(u_{0,h}\) and Corollary 2.4. Using (3.30), it follows
\(\square \)
3.3 Main result
Theorem 3.3
Let \(f_h\), \(\tilde{v}_0\), \(\tilde{v}_1\), \(\tilde{u}_{j,h}\), \(j=0,1,2\) and \(\tilde{u}_h\) be given as above. Then there exists some \(h_0\in (0,1]\) such that for \(h\in (0,h_0]\) there are initial values \((u_{0,h}, u_{1,h})\) satisfying (A.1)–(A.3) and such that
Moreover, if \(u_h\) solves (3.1)–(3.4), then
Proof
Given \((u_{3,h}, u_{4,h})\) we construct \((u_{0,h}, u_{1,h}, u_{2,h})\) such that (A.1)–(A.3) holds. First we note that \(\Vert u_{4,h}\Vert _{L^2(\Omega )}\) is of order \(h^2\) as \(\partial _{x_1}^l v^{4}\) is bounded in \(L^2(0,L)\) for \(l=0,1\). Moreover, we have
for \(j=1,2\) and
Using the structure of \(u_{3,h}\) we obtain
Altogether we obtain that \(u_{3,h}\) and \(u_{4,h}\) satisfy (A.1)–(A.3), the necessary conditions for the large times existence result in the appendix. The assumptions on g and the structure of \(f_h\) imply that (A.4) and (A.5) are fulfilled. Applying Lemma 3.1 and 3.2, we obtain for \(h_0\) sufficiently small, the existence of \((u_{0,h}, u_{1,h}, {\bar{u}}_{2,h})\) such that
and
for all \(\varphi \in \mathcal {B}\). We use the ansatz \(u_{2,h} = {\bar{u}}_{2,h} + \gamma _2^h x^\perp \) and \({\bar{u}}_{2+j,h} = u_{2+j,h} + \gamma _{2+j}^h x^\perp \) for \(j=1,2\). Choosing
it follows
for all \(\varphi \in H^1_{\textrm{per}}(\Omega ;\mathbb {R}^3)\). Moreover, for
we deduce
for all \(\varphi \in H^1_{\textrm{per}}(\Omega ;\mathbb {R}^3)\). Then it holds \(|\gamma _2^h|\le C h^2\) as
and \(|\gamma _3^h| \le C h^2\) with a similar calculation. Lastly, we need to find \(\gamma _4^h\) such that
for all \(\varphi \in H^1_{\textrm{per}}(\Omega ;\mathbb {R}^3)\). Therefore, we choose
The first and last term can be bounded easily, using Corollary 2.4
and
For the second part of \(\gamma _4^h\), we use the following equality
where
as \(\Vert u_{0,h}\Vert _{H^1_h(\Omega )} \le Ch^2\) and \(|P|_h = |P|\), because \(P\in \mathbb {R}^{3\times 3}_{skew}\). Furthermore, we obtain with (2.5)
Utilizing the inequality for the initial values (3.24), we deduce
Lastly due to the symmetry properties of \(D^3\tilde{W}\), the structure of \(\nabla _h \tilde{u}_{0,h}\) and (2.5), it follows
where
Due to the structure of Q and \(P = \nabla x^\perp \), it follows
Hence, with \(R = O(h^4)\) we obtain
Thus, altogether, it follows with \(|\gamma _2^h| \le Ch^2\)
We obtain for \(h_0\) sufficiently small, the existence of \((u_{0,h}, u_{1,h}, u_{2,h})\) such that (A.1)–(A.3) are satisfied and
holds.
Due to Theorem A.1, there exists a solution \(u_h\in \bigcap _{k=0}^4 C^k([0,T]; H^{4-k}_{\textrm{per}}(\Omega ;\mathbb {R}^3))\) of (3.1)–(3.4). Thus, \(w_h := u_h - \tilde{u}_h\) solves the system
for all \(\varphi \in C^1([0,T]; H^1_{\textrm{per}, (0)}(\Omega ;\mathbb {R}^3))\) with \(\varphi |_{t=T} = 0\) and with \(w_{j,h} := u_{j, h} - \tilde{u}_{j,h}\), \(j=0,1\). Hence, with (A.16) we obtain an upper bound for w. For this we use that, due to the structure of \(r_h\) and \(r_{N,h}\), it follows
as a, b, c and v are sufficiently regular. Moreover, using (3.24)
for \(k=0,1\), where we used Poincaré’s and Korn’s inequality, as well as the fact that \(w_{k,h}\in \mathcal {B}\) holds for \(k=0,1\). With the fundamental theorem of calculus and Corollary 2.4, we deduce
Lastly, we have to deal with the rotational term. Using the momentum balance law, \(u_{0,h}\), \(u_{1,h}\in \mathcal {B}\) and the structure of g, we obtain with \(q^h := h^2 f^h\)
Hence, it follows
as due to (A.6)
for \(\delta = 0,2\). Thus, with (A.16) it follows
\(\square \)
Change history
02 November 2022
Missing Open Access funding information has been added in the Funding Note.
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Acknowledgements
Tobias Ameismeier was supported by the RTG 2339 “Interfaces, Complex Structures, and Singular Limits” of the German Science Foundation (DFG). The support is gratefully acknowledged.
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Appendix A: Large times existence for the non-linear problem
Appendix A: Large times existence for the non-linear problem
The existence of solutions follows from
Theorem A.1
Let \(\theta \ge 1\), \(0< T< \infty \), \(f_h\in W^3_1(0,T; L^2(\Omega )) \cap W^1_1(0,T; H^2_{\textrm{per}}(\Omega ))\), \(h\in (0,1]\) and \(u_{0,h}\in H^4_{\textrm{per}}(\Omega )\), \(u_{1,h}\in H^3_{\textrm{per}}(\Omega )\) such that
where
Moreover, we assume for the initial data
and for the right hand side
uniformly in \(0<h\le 1\). Then there exists \(h_0\in (0,1]\) and \(C>0\) depending only on M and T such that for every \(h\in (0,h_0]\) there is a unique solution \(u_h\in \bigcap _{k=0}^4 C^k([0,T]; H^{4-k}_{\textrm{per}}(\Omega ))\) of (3.1)–(3.4) satisfying
uniformly in \(0<h\le h_0\).
Proof
A proof can be found in [4, Theorem 5.1.1] or [1]. \(\square \)
The linearized system for (3.1)–(3.4) is given by
We want to show h-independent estimates for solutions of the linearized system. For this we assume that \(u_h\) satisfies for \(0 < h \le 1\)
where \(R\in (0,R_0]\), with \(R_0\) chosen later appropriately small.
Lemma A.2
Assume that (A.11) holds, \(t\in [0, T]\) and \(0< R\le R_0\). Then
for \(1 \le |\beta | \le 3\).
Proof
For a proof see [4, Lemma 5.2.2]. \(\square \)
For the higher regularity estimates, we need the following result.
Theorem A.3
Assume \(u_h\) satisfies (A.11). Then there exist \(C>0\) and \(R_0\in (0,1]\) such that if \(\varphi \in H^{2+k}_{\textrm{per}}(\Omega )\) solves for some \(g\in H^k_{\textrm{per}}(\Omega )\) and \(g_N\in L^2(0,L;H^{k+\frac{1}{2}}(\partial S))\cap H^{k}(0,L; H^{\frac{1}{2}}(\partial S))\)
then
Proof
See [1, Theorem 3.5]. \(\square \)
In order to bound differences between the approximation \(\tilde{u}_h\) and the analytic solution \(u_h\) we consider the following weak form of the linearized system (A.7)–(A.10):
for all \(\varphi \in C^1([0,T]; H^1_{\textrm{per}, (0)}(\Omega ;\mathbb {R}^3))\) with \(\varphi |_{t=T} = 0\). Here we denote \(Q_T := \Omega \times (0,T)\) and
equipped with the h-dependent norm
Lemma A.4
Assume that \(u_h\) satisfies (A.11) with \(R\in (0,R_0]\) and \(h\in (0,1]\). Let \(R_0\) be sufficiently small and \(w\in C^0([0, T]; X_h)\cap C^1([0,T];L^2(\Omega ;\mathbb {R}^3))\) be a solution of (A.15) for \(f_1\in L^1(0,T;L^2(\Omega , \mathbb {R}^{3\times 3}))\), \(f_2 \in L^1(0,T;L^2(\Omega ;\mathbb {R}^3))\), \(a_N\in L^1(0,T; H^1(\Omega ;\mathbb {R}^3))\) \(w_0\in L^2(\Omega ;\mathbb {R}^3)\) and \(w_1\in X'_h\). Then there are \(C_0\), \(C>0\) independent of w and T such that
where \(u(t) := \int \limits _0^t w(\tau ) d\tau \) and \((L^2_h)'\) is an abbreviation for \((L^2_h(\Omega ;\mathbb {R}^{3\times 3}))'\).
Proof
Let \(0 \le T' \le T\) and define \(\tilde{u}_{T'}(t) = -\int \limits _t^{T'} w(\tau ) d\tau \). We use, after smooth approximation, \(\varphi = \tilde{u}_{T'} \chi _{[0,T']}\). Then it follows
Using
it follows with \(\tilde{u}_{T'}(0) = -u(T')\)
where we used Lemma A.2 and Korn’s inequality, as well as the subsequent inequalities
Now we can use \(\tilde{u}_{T'}(0) = -u(T')\) and \(\tilde{u}_{T'}(t) = -u(T') + u(t)\) to deduce
Using the later inequalities and applying the supremum over \(T'\in [0, {\bar{T}}]\) such that \(R{\bar{T}} \le \kappa \), \(\kappa \in (0,1]\), it follows
Hence, with Young’s inequality and \(\kappa \), thus \({\bar{T}}\), small enough, we can conclude with an absorption argument that
Applying now the Lemma of Gronwall, we obtain (A.16) for all \(0< T < \infty \) such that \(RT \le \kappa \) holds.
For an arbitrary \(0< T <\infty \), we choose \(0=T_0< T_1< \cdots< T_{N-1} < T_N = T\) such that \(\frac{1}{2}\kappa \le R(T_{j+1} - T_j) \le \kappa \) for \(j=0,\ldots N-1\). Then we use \(\varphi = \tilde{u}_{T_{j+1}} \chi _{[T_{j},T_{j+1}]}\) and obtain via analogous arguments as above, because of \(R(T_{j+1} - T_j) \le \kappa \),
Hence, an iterative application leads to
Finally, due to \(\frac{1}{2}\kappa \le R(T_{j+1} - T_j)\), we obtain \(N\le 2\kappa ^{-1}RT\) and thus
Hence, (A.16) holds for some \(C_0\), \(C >0\) independent of \(R\in (0,R_0]\), \(h\in (0, 1]\) and \(0< T <\infty \). \(\square \)
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Abels, H., Ameismeier, T. Convergence of thin vibrating rods to a linear beam equation. Z. Angew. Math. Phys. 73, 166 (2022). https://doi.org/10.1007/s00033-022-01803-y
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DOI: https://doi.org/10.1007/s00033-022-01803-y