Abstract
This article develops for the first time a rigorous analysis of Hibler’s model of sea ice dynamics. Identifying Hibler’s ice stress as a quasilinear second-order operator and regarding Hibler’s model as a quasilinear evolution equation, it is shown that a regularized version of Hibler’s coupled sea ice model, i.e., the model coupling velocity, thickness and compactness of sea ice, is locally strongly well-posed within the \(L_q\)-setting and also globally strongly well-posed for initial data close to constant equilibria.
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1 Introduction
Sea ice is a material with a complex mechanical and thermodynamical behavior. Freezing sea water forms a composite of pure ice, liquid brine, air pockets and solid salt. The details of this formation depend on the laminar or turbulent environmental conditions, see e.g., Hibler (1979), Feltham (2008) and Golden (2015). This composite responds differently to heating, pressure or mechanical forces than for example the (salt-free) glacial ice of ice sheets.
The evolution of sea ice has attracted much attention in climate science due to its role as a hot spot in global warming. The state of the art concerning the modeling of sea ice is described in the very recent survey article (Golden et al. 2020) in the Notices of the American Mathematical Society.
Somewhat surprisingly, the field of sea ice dynamics forms a terra incognita to rigorous mathematical analysis. In contrast to atmospheric or oceanic models, see e.g., the work of Lions et al. (1992a, 1992b) and Cao and Titi (2007) for the primitive equations as well as the work of Majda (2003) and Benacchio and Klein (2019) for atmospheric flows, rigorous analysis of sea ice models is essentially non-existent.
The governing equations of large-scale sea ice dynamics that form the basis of virtually all sea ice models in climate science are suggested in a seminal paper by Hibler (1979). Sea ice is here modeled as a material with a very specific constitutive law based on viscous–plastic rheology. This model has been investigated numerically by various communities (see e.g., Mehlmann et al. 2021; Mehlmann and Korn 2021; Mehlmann 2019; Mehlmann and Richter 2017; Seinen and Khouider 2018; Danilov et al. 2015; Kimmrich et al. 2015; Bouchat and Tremblay 2014; Lemieux and Tremblay 2009), but it seems it never was studied from a rigorous analytical point of view. In fact, even the existence of weak solutions to Hibler’s sea ice model seems to be unknown until today.
Moreover, fundamental questions in this respect such as thermodynamical consistency of the Hibler model with the second law of thermodynamics, as well as existence, uniqueness and regularity properties of solutions of this sea ice PDE system, seem to be open problems. Under distinct simplifications, certain submodels were, however, considered in Gray (1999) and Guba et al. (2013) within the context of hyperbolic systems. The authors postulate ill-posedness of these simplified submodels.
As stated above, Hibler’s sea ice model was already investigated numerically by many authors. All of these approaches are based on various regularizations of the underlying ice stress tensor. For example, the original viscous–plastic equations have been regularized by means of additional artificial elasticity, see e.g., the work of Hunke–Dukowicz (1997), in order to improve computational efficiency. This elastic–viscous–plastic approach has been implemented then in many sea ice and climate models. Let us emphasize that simulations of sea ice behavior show a distinct discrepancy whether the original Hibler or the regularized sea ice PDEs are being used, see Losch and Danilov (2012). For a thorough numerical study based on a regularization of the original viscous–plastic ice stress, we refer to the work of Mehlmann and Korn (2021), Mehlmann and Richter (2017) and Mehlmann (2019).
The modeling of sea ice dynamics is a very active and dynamic field of research (Golden et al. 2020). In particular, we refer to alternative models e.g., by Moritz and Ukita (2000), Wilchinsky and Feltham (2004) and Eisenman and Wettlaufer (2009) as well as to the ones described in the survey paper (Golden et al. 2020).
In this article for the first time, we rigorously prove the existence and uniqueness of a strong solution to Hibler’s sea ice model. Our approach is based on the theory of quasilinear parabolic evolution equations and a regularization of Hibler’s original ice stress \(\sigma \). This regularization has been used already in various numerical approaches by Mehlmann, Richter and Korn, see Mehlmann (2019), Mehlmann and Richter (2017) and Mehlmann and Korn (2021).
A key point of our analysis is the understanding of the term \({{\,\mathrm{div}\,}}\sigma \) as a strongly elliptic quasilinear operator A within the \(L_q\)-setting. We show that its linearization, subject to Dirichlet boundary conditions, fulfills the Lopatinskii–Shapiro condition yielding the maximal \(L_q\)-regularity property of the linearized Hibler operator. The latter property is then extended to the coupled system, described precisely later on in (2.10), consisting of the momentum equation for the velocity u and the two balance laws for the mean ice thickness h and the ice compactness a. Regarding this model as a quasilinear evolution equation, we obtain strong well-posedness of the fully coupled system. For background information on linear and quasilinear evolution equations, we refer to Arendt et al. (2011), Denk et al. (2003, 2004, 2007), Kunstmann and Weis (2004), Prüss and Simonett (2016) and Hieber et al. (2020).
In our first main result, we prove the existence and uniqueness of a local strong solution to (2.10) for suitably chosen initial data and show that this solution depends continuously on the data, exists on a maximal time interval and regularizes instantly in time. Secondly, we show that this solution extends uniquely to a global strong solution provided the initial data are close to the equilibria \(v_* = (0,h_*,a_*)\), where \(h_*\) and \(a_*\) denote constants and the external forces vanish.
To put our result in perspective, note that the existence of a weak solution to Hibler’s sea ice model (2.10) is not known until today. It is also interesting to observe that Hibler’s sea ice stress tensor is related to the stress tensor of certain non-Newtonian fluids, as described in Bothe and Prüss (2007). It was shown recently by Burczak et al. (2021) that under certain assumptions weak solutions to these non-Newtonian fluid models are highly non unique.
This article is organized as follows: Sect. 2 presents Hibler’s model as well as our main well-posedness results for this system. In Sect. 3, we rewrite Hibler’s operator as a second-order quasilinear operator, whose linearization will be investigated in Sect. 4. There we show that the linearization of Hibler’s operator is a strongly elliptic operator within the \(L_q\)-setting and that this operator subject to Dirichlet boundary conditions satisfies the maximal \(L_q\)-regularity property. After a short section on functional analytic properties of Hibler’s operator, in Sect. 6, we present the proof of our local well-posedness result. Finally, the proof of our global well-posedness result is given in Sect. 7.
After having finished our article, we became aware of the work by Liu et al. (2021), which studies a related problem.
2 Hibler’s Viscous–Plastic Sea Ice Model and Main Results
Hibler (1979) proposed a rheology model for sea ice dynamics, which has become since then the standard sea ice dynamics model and serves until today as a basis for many numerical studies in this field. Roughly speaking, pack ice consists of rigid plates which drift freely in open water or are closely packed together in areas of high ice concentration. Although individual ice floes may have very different sizes, pack ice may be considered as a highly fractured two-dimensional continuum.
The momentum balance in this model is given by the two-dimensional equation:
where \({u: (0,\infty ) \times {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2}\) denotes the horizontal ice velocity and m the ice mass per unit area. Moreover, \({- m c_{{\mathrm{cor}}}\, n \times u}\) represents the Coriolis force with Coriolis parameter \(c_{{\mathrm{cor}}}> 0\) and unit vector \({n : {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^3}\) normal to the surface, while \(- m g \nabla H\) describes the force arising from changing sea surface tilt with sea surface dynamic height \({H: (0,\infty ) \times {\mathbb {R}}^2 \rightarrow [0,\infty )}\) and gravity g. The terms \(\tau _{\mathrm{{atm}}}\) and \(\tau _{\mathrm{{ocean}}}\) describe atmospheric wind and oceanic forces given by
where \(U_{\mathrm{{atm}}}\) and \(U_{\mathrm{{ocean}}}\) denote the velocity of the surface winds and current, respectively. Furthermore, \(C_{\mathrm{{atm}}}\) and \(C_{\mathrm{{ocean}}}\) are air and ocean drag coefficients, \(\rho _{\mathrm{{atm}}}\) and \(\rho _{\mathrm{{ocean}}}\) denote the densities for air and sea water, and \(R_{\mathrm{{atm}}}\) and \(R_{\mathrm{{ocean}}}\) are rotation matrices acting on wind and current vectors. For results on fluids driven by wind forces, we refer to Bresch and Simon (2001).
Following Hibler (1979), the viscous–plastic rheology is given by a constitutive law that relates the internal ice stress \(\sigma \) and the deformation tensor \({\varepsilon = \varepsilon (u) = \frac{1}{2}(\nabla u + \nabla u ^\mathrm{T})}\) through an internal ice pressure P and nonlinear bulk and shear viscosities, \(\zeta \) and \(\eta \), such that the principal components of the stress lie on an elliptical yield curve with the ratio of major to minor axes e. This constitutive law is given by
The pressure P measures the ice strength, depending on the thickness h and the ratio a of thick ice per unit area, and is explicitly given by
where \(p^*>0\) and \(c>0\) are given constants. The bulk and shear viscosities \(\zeta \) and \(\eta \) increase with pressure and decreasing deformation tensor and are given by
where
and e as described above is the ratio of the long axis to the short axis of the elliptical yield curve. The above law represents an idealized viscous–plastic material, whose viscosities, however, become singular if \(\triangle \) tends to zero.
For this reason, already Hibler proposed to regularize this behavior by bounding the viscosities when \(\triangle \) is getting small and by defining maximum values \(\zeta _{\mathrm{max}}\) and \(\eta _{\mathrm{max}}\) for \(\zeta \) and \(\eta \). Then, \(\zeta \) and \(\eta \) become
This formulation of the viscosities leads, however, to non-smooth rheology terms. To enforce smoothness, several regularizations have been considered in the literature, see e.g., Mehlmann and Richter (2017), Lemieux and Tremblay (2009). For example, Lemieux and Tremblay (2009) replaced \(\zeta \) by \(\zeta = \zeta _{\hbox {\tiny max}} \tanh (P/(2\triangle \zeta _{\mathrm{max}})\).
An elastic–viscous–plastic stress tensor was introduced by Hunke and Dukowicz (1997). Starting from the observation that the relation (2.4) for \(\sigma \) can be rewritten as \(\frac{1}{2\eta }\sigma + \frac{\eta -\zeta }{4 \eta \zeta } {{\,\mathrm{tr}\,}}\sigma + \frac{P}{4\zeta }I = \varepsilon \), they proposed the relation
Note that the relation (2.4) is obtained in the limit \(E \rightarrow \infty \), while for \(\zeta ,\eta \rightarrow \infty \), one recovers the elasticity equation \(\frac{1}{E}\partial _t \sigma = \varepsilon \). In the pure plastic case, the compressive stress \(\sigma _d = {{\,\mathrm{tr}\,}}\sigma \) and the shear stress \(\sigma _s= ( (\sigma _{11}-\sigma _{22})^2 + 4 \sigma _{12}^2)^{1/2}\) are linked by the relation \((\sigma _d + P)^2 + {e}^2\sigma _s^2 = P^2\), which leads to an elliptical yield curve.
Following Mehlmann and Korn (2021), see also Kreyscher et al. (2000), we consider for \(\delta >0\) the regularization
We then set \(\zeta _\delta =\frac{P}{2 \triangle _\delta (\varepsilon )}\) and \(\eta _\delta ={e}^{-2} \zeta _\delta \) as well as
We consider in the following the above momentum equation (2.1) in a bounded domain \(\Omega \subset {\mathbb {R}}^2\) with boundary of class \(C^2\). It is coupled to two balance equations for the mean ice thickness
and the ice compactness \(a: J \times \Omega \rightarrow {\mathbb {R}}\) with \(a \ge \alpha \) for some \(\alpha \in {\mathbb {R}}\) given by
Here, \(\kappa \) is a small parameter that indicates the transition to open water in the sense that for \(m = \rho _{\mathrm{{ice}}}h\) a value of h(t, x, y) less than \(\kappa \) means that at \((x,y) \in \Omega \) and at time t there is open water. Furthermore, let \(J = (0,T)\) for \(0<T\le \infty \), let \(\Delta \) be the Laplacian, \(d_h>0\) and \(d_a>0\) be constants and for an arbitrary \(f \in C^1([0,\infty );{\mathbb {R}})\), for example the one considered by Hibler (1979), define the terms \(S_h\) and \(S_a\) by
The system is finally completed by Dirichlet boundary conditions for u and Neumann boundary conditions for h and a.
Given \(0<T\le \infty \) and \(J= (0,T)\), the complete set of equations describing sea ice dynamics by Hibler’s model then reads as
Note that \(m = \rho _{\mathrm{{ice}}}h\) and h is subject to (2.7).
To formulate our main well-posedness result for the system (2.10), we first rewrite it as a quasilinear evolution equation and introduce a setting as follows. Denoting the principle variable of the system by \(v=(u,h,a)\), we rewrite system (2.10) as a quasilinear evolution equation of the form
Here, v belongs to the ground space \(X_0\) defined by
where \(1<q<\infty \). The regularity space will be
Furthermore, the quasilinear operator A(v) is given by the upper triangular matrix
Here, \(A^\mathrm{H}_\mathrm{D}\) denotes the realization of Hibler’s operator subject to Dirichlet boundary conditions on \(L_q(\Omega ;{\mathbb {R}}^2)\), introduced and defined precisely in Sect. 3, and \(\Delta _\mathrm{N}\) the Neumann Laplacian on \(L_q(\Omega )\) defined by \((\Delta _\mathrm {N} = \Delta )\) \(D(\Delta _\mathrm {N}) = \{h \in H^{2}_q(\Omega ): \partial _\nu h =0 \text{ on } \partial \Omega \}\). The semilinear part F(v) is defined by
where \(c_1 = \rho _{\mathrm{{atm}}}C_{\mathrm{{atm}}}R_{\mathrm{{atm}}}\rho _{\mathrm{{ice}}}^{-1}\) and \(c_2 = \rho _{\mathrm{{ocean}}}C_{\mathrm{{ocean}}}R_{\mathrm{{ocean}}}\rho _{\mathrm{{ice}}}^{-1}\) and \(U_{\mathrm{{atm}}}\) and \(U_{\mathrm{{ocean}}}\) are given functions.
We consider solutions v within the class
where \(J=(0,T)\) as above is an interval and \(\mu \in (1/p,1]\) indicates a time weight. More precisely,
The time trace space of this class is given by
provided \(p \in (1,\infty )\) and \(\mu \in (1/p,1]\). Note that [see e.g., Section 7 of Adams and Fournier (2003)]
provided
It is well-known that for \(\mu \) satisfying (2.16), the above real interpolation spaces can be characterized as
For brevity, we set \(X_\gamma :=X_{\gamma ,1}\). Moreover, let \(V_\mu \) be an open subset of \(X_{\gamma ,\mu }\) such that all
Theorem 2.1
(Well-posedness of Hibler’s sea ice model)
Let \(\Omega \subset {\mathbb {R}}^2\) be a bounded domain with boundary of class \(C^{2}\), and for \(\delta >0\), let \(\sigma _\delta \) be defined as in (2.6). Assume that \(1<p,q <\infty \) and that \(\mu \in (1/p,1]\) are subject to (2.16) and let \(v_0 \in V_\mu \), where \(V_\mu \) is as in (2.17).
(a) Then there exist \(\tau =\tau (v_0)>0\) and \(r = r(v_0)>0\) with \({\overline{B}}_{X_{\gamma ,\mu }}(v_0,r) \subset V_\mu \) such that Eq. (2.11), i.e., Eqs. (2.10), (2.2), (2.3), (2.5), (2.8) and (2.9), has a unique solution
for each initial value \(v_1 \in {\overline{B}}_{X_{\gamma ,\mu }}(v_0,r)\). Moreover, there exists \(C=C(v_0)\) such that
for all \(v_1,v_2 \in {\overline{B}}_{X_{\gamma ,\mu }}(v_0,r)\). In addition,
i.e., the solution regularizes instantly in time. In particular,
for any \(b \in (0,\tau )\).
(b) The solution \(v= v(v_0)\) exists on a maximal time interval \(J(v_0) = [0,t^+(v_0))\), which is characterized by the following alternatives:
-
(i)
global existence, i.e., \(t_+(v_0) = \infty \),
-
(ii)
\(\lim _{t \rightarrow t_+(v_0)} \text {dist}_{X_{\gamma ,\mu }} \big (v(t),\partial V_\mu \big ) =0\),
-
(iii)
\(\lim _{t \rightarrow t_+(v_0)} v(t)\) does not exist in \(X_{\gamma ,\mu }\).
Remark 2.2
(a) Assuming \(1<p,q<\infty \) and \(\mu \in (1/p,1]\) subject to (2.16), the smoothness condition required for the initial data \(v_0=(u_0,h_0,a_0)\) in Theorem 2.1 can be characterized as
(b) These conditions are in particular satisfied if \((u_0, h_0, a_0) \in H^{1+2/q+s}_q(\Omega )^4\) for \(s>0\) satisfy the above boundary conditions.
Assuming that \(h_*\) and \(a_*\) are constant in time and space, we observe that \((0,h_*,a_*)\) are trivial equilibria for Eq. (2.11) subject to vanishing forcing terms. We proceed by showing that the equilibrium \((0,h_*,a_*)\) is stable in \(X_{\gamma ,\mu }\), and the unique solution of (2.11) exists globally for initial data close to the aforementioned equilibrium provided \(\delta \) is chosen small enough and the external forces vanish.
Theorem 2.3
There exists \(\delta ^* > 0\) such that for all \(\delta \in (0,\delta ^*)\) and for \(h_*\) and \(a_*\) as above, the equilibrium \(v_*= (0, h_*, a_*)\) is stable in \(X_{\gamma ,\mu }\), and there exists \(r > 0\) such that the unique solution v of (2.11) without forcing terms and with initial value \(v_0 \in X_{\gamma ,\mu }\) fulfilling \(\Vert v_0 - v_*\Vert _{X_{\gamma ,\mu }} < r\) exists on \({\mathbb {R}}_+\) and converges at an exponential rate in \(X_{\gamma ,\mu }\) to some equilibrium \(v_\infty \) of (2.11) as \(t \rightarrow \infty \).
Remark 2.4
Equation (2.10) shows that without forcing, the solution tends to the equilibria \(h_*= \frac{1}{|\Omega |}\int _{\Omega } h(0) \hbox {d}x\) and \(a_*= \frac{1}{|\Omega |}\int _\Omega a(0) \hbox {d}x\) determined by the initial mean values of h and a, respectively.
3 Hibler’s Ice Stress Viewed as a Second Order Quasilinear Operator
In this section, we interpret the term \({{\,\mathrm{div}\,}}\sigma \) as a quasilinear second-order operator. To this end, denote by \(\varepsilon =(\varepsilon )_{ij}\) the deformation or rate of strain tensor and define the map \({\mathbb {S}} :{\mathbb {R}}^{2\times 2} \rightarrow {\mathbb {R}}^{2\times 2}\) in such a way that
If \(\varepsilon \in {\mathbb {R}}^{2\times 2}\) is identified with the vector \((\varepsilon _{11}, \varepsilon _{12}, \varepsilon _{21}, \varepsilon _{22})^\mathrm{T} \in {\mathbb {R}}^4\), \({\mathbb {S}}\) corresponds to the symmetric positive semi-definite matrix
and we obtain
The stress tensor \(\sigma =\sigma (\varepsilon ,P)\) can then be represented as
As explained in Sect. 2, for \(\delta >0\), we then substitute S by
and define Hibler’s operator as
Employing product and chain rule as well as symmetries of \({\mathbb {S}}\), we infer that
for \(i=1,2\). Exploiting symmetries of \({\mathbb {S}}\) and \(\varepsilon \) once again, we conclude that
for \(i=1,2\) and \(D_m = - \mathrm {i} \partial _m\).
We denote the coefficients of the principal part of \(\mathcal {A}^\mathrm{H}\) by
In view of the symmetries of \({\mathbb {S}}\) and \({\mathbb {S}} \varepsilon \), we conclude that
For given \(v_0=(u_0, h_0, a_0) \in V_\mu \) with \(\mu > \frac{1}{2} +\frac{1}{p} + \frac{1}{q}\), let
be Hibler’s operator with frozen coefficients. The representation in (3.5) shows that the principal coefficients \(a^{kl}_{ij}(\nabla u_0,P(h_0,a_0))\) of \(\mathcal {A}^\mathrm{H}(v_0)\), as well as lower-order terms, depend smoothly on \(u_0\), \(h_0\) and \(a_0\) with respect to the \(C^1\)-norm. Moreover, the embedding (2.15) yields that they lie in \(C({\overline{\Omega }})\).
4 Hibler’s Operator: Ellipticity and Maximal Regularity
In this section, we show that Hibler’s operator \(\mathcal {A}^\mathrm{H}(v_0)\) given as in (3.7) defines a strongly elliptic operator and, when subject to Dirichlet boundary conditions, satisfies the Lopatinskii–Shapiro condition. This implies then that the \(L_q\)-realization \(A^\mathrm{H}_\mathrm{D}(v_0)\) of \(\mathcal {A}^\mathrm{H}(v_0)\) given by
satisfies the maximal \(L_q\)-regularity property and furthermore that \(A^\mathrm{H}_\mathrm{D}(v_0)\) admits a bounded \(H^\infty \)-calculus on \(L_q(\Omega ;{\mathbb {R}}^2)\).
For \(\theta \in (0,\pi ]\), let \({\Sigma _\theta := \{z \in {\mathbb {C}}\setminus \{0\} : \vert \arg z \vert < \theta \}}\) be a sector in the complex plane, and let \(D=-\mathrm {i}(\partial _1,\ldots ,\partial _n)\). We start by recalling from Denk et al. (2003) that for \(x \in {\mathbb {R}}^n\) an operator \({{\mathcal {B}}}\) of the form \({{\mathcal {B}}}(x,D)=\sum _{|\alpha |\le 2} b_\alpha (x)D^\alpha \) with continuous top-order coefficients \(b_\alpha \in {{\mathcal {L}}}(E)\), E an arbitrary Banach space, is said to be parameter–elliptic of angle \(\phi \in (0,\pi ]\) if the spectrum \({\sigma ({{\mathcal {B}}}_{\#}(x,\xi ))}\) of the symbol of the principal part \({{\mathcal {B}}}_{\#}(x,\xi ) = \sum _{|\alpha |=2} b_\alpha (x)\xi ^{\alpha }\) satisfies
for every \(x \in {\mathbb {R}}^n\) and every \(\xi \in {\mathbb {R}}^n\) with \(\vert \xi \vert = 1\). We call \(\phi _{\mathcal {B}} = \inf \{\phi : \sigma (\mathcal {B}_{\#}(x,\xi )) \subset \Sigma _\phi \}\) the angle of ellipticity of \({{\mathcal {B}}}\). Moreover, the operator \({{\mathcal {B}}}(x,D)\) is called normally elliptic if it is parameter–elliptic of angle \(\phi _{{{\mathcal {B}}}} < \pi /2\). If E is a Hilbert space, an operator \({{\mathcal {B}}}\) of the above form \({{\mathcal {B}}}(x,D)=\sum _{|\alpha |= 2} b_\alpha (x)D^\alpha \) is called strongly elliptic if there exists a constant \(c > 0\) such that
for all \(x \in {\mathbb {R}}^n\), \(\xi \in {\mathbb {R}}^n\) with \(\vert \xi \vert = 1\) and all \(w \in E\). Here, \((\cdot \vert \cdot )_E\) denotes the inner product on E. To understand this condition, let n(L) be the numerical range of a bounded linear operator on E, i.e., n(L) is the closure of the set consisting of all \(z \in {\mathbb {C}}\) such that \(z=(Lw \vert w)_E\) for some \(w \in E\) with \(\Vert w \Vert _E=1\). Since \(\sigma (L) \subset n(L)\), we see that every strongly elliptic operator \({{\mathcal {B}}}\) is parameter–elliptic of angle \(\phi _{{\mathcal {B}}}< \pi /2\), hence even normally elliptic.
Consider now the special case of homogeneous differential operators acting on \({\mathbb {C}}^n\)-valued functions as
Here, \(\Omega \subset {\mathbb {R}}^n\) denotes a domain with boundary of class \(C^2\). Its symbol is defined as
We now show that \(\mathcal {A}^\mathrm{H}(v_0)\) is strongly elliptic provided \(v_0 \in V_\mu \).
Proposition 4.1
Let \(p,q \in (1,\infty )\) and \(\mu \in (\frac{1}{p},1]\) such that (2.16) holds. Then, for fixed \(v_0 \in V_\mu \), the principal part of Hibler’s operator \(\mathcal {A}^\mathrm{H}(v_0)\) defined as in (3.7) is strongly elliptic, and moreover parameter–elliptic of angle \(\phi _{\mathcal {A}^\mathrm{H}(v_0)} =0\).
Proof
Recall that the principal part of \(\mathcal {A}^\mathrm{H}(v_0)\) is given by
with
as in (3.5) and \(P_0=P(h_0,a_0)\). Taking into account the underlying symmetries, we see that the symbol of the principal part of \(\mathcal {A}^\mathrm{H}(v_0)\) is given by
For given \(d \in {\mathbb {R}}^{2\times 2}\), we use the notation
To verify condition (4.2), first recall that for any given \(d \in {\mathbb {R}}^{2\times 2}\), [see (3.1)],
Furthermore, using Young’s inequality, we estimate
Due to our assumptions on \(v_0\), the function \(\frac{P_0}{2\triangle _\delta (\varepsilon )^3}\) is real-valued, bounded, continuous and positively bounded from below by a constant \(c_{\delta ,\kappa ,\alpha } > 0\). Thus, combining (4.4) and (4.5), for all \(d \in {\mathbb {R}}^{2\times 2}\), we obtain
We can now verify condition (4.2). Given \(\xi \in {\mathbb {R}}^2\) and \(\eta \in {\mathbb {C}}^2\) with \(|\xi | = |\eta | =1\), set \(\eta _i =: x_i + \mathrm {i} y_i\) for \(i = 1,2\). Because of the symmetries of \((a^{ij}_{kl})\) as pointed out in (3.6) and using (4.6), we derive
Moreover, using \(|\xi | = 1\),
so using \(|\eta | = 1\),
and thus \(\mathcal {A}^\mathrm{H}(v_0)\) is strongly elliptic with an ellipticity constant \(\ge \frac{c_{\delta ,\kappa ,\alpha } \delta }{{e}^2} \).
To prove parameter–ellipticity of \(\mathcal {A}^\mathrm{H}(v_0)\), we first note that due to (4.3), strong ellipticity implies normal ellipticity, and by symmetry of \(\mathcal {A}_{\#}^\mathrm{H}\), we conclude that \({\sigma (\mathcal {A}_{\#}^\mathrm{H})(x,\xi ) \subset {\mathbb {R}}_{+}}\) is valid for every \({x \in {\overline{\Omega }}}\) and \({\xi \in {\mathbb {R}}^2}\) with \({\vert \xi \vert = 1}\). This implies parameter–ellipticity of \(\mathcal {A}^\mathrm{H}(v_0)\) with \({\phi _{\mathcal {A}^\mathrm{H}(v_0)} = 0}\). \(\square \)
The assertion of the following lemma will be crucial in the proof of the fact that the linearized Hibler operator \(\mathcal {A}^\mathrm{H}(v_0)\) subject to Dirichlet boundary conditions satisfies the Lopatinskii–Shapiro condition.
Lemma 4.2
Let \(p,q \in (1,\infty )\) and \(\mu \in (\frac{1}{p},1]\) such that (2.16) holds. For fixed \(v_0 \in V_\mu \), let \(a_{ij}^{kl}\) be the coefficients of the principal part of Hibler’s operator \(\mathcal {A}^\mathrm{H}(v_0)\) defined as in (3.5). Assume that \({x \in \partial \Omega }\), \(\xi \), \({\nu \in {\mathbb {R}}^2}\) with \({\vert \xi \vert = \vert \nu \vert = 1}\) and \({(\xi \vert \nu ) = 0}\) as well as u, \({v \in {\mathbb {C}}^2}\). Then,
Proof
Let \({x \in \partial \Omega }\), \(\xi \), \({\nu \in {\mathbb {R}}^2}\) with \({\vert \xi \vert = \vert \nu \vert = 1}\) and \({(\xi \vert \nu ) = 0}\) as well as u, \({v \in {\mathbb {C}}^2}\). We introduce the notation \({u_i = x_i + \mathrm {i} y_i}\) and \({v_i = {\tilde{x}}_i + \mathrm {i} {\tilde{y}}_i}\), \({i=1,2}\). Using the symmetries of \((a^{kl}_{ij})\) as in (3.6) and the estimate (4.6), we obtain
Thus, condition (4.7) is satisfied. To verify condition (4.8), it remains to consider the case \( = 0\) in the last line and deduce
For general \(d \in {\mathbb {R}}^{2\times 2}\), \(\triangle ^2(d) = 0\) implies \(d_{11} = d_{22} = 0\), so from \( = 0\) in (4.9), we obtain
Due to \(|\xi | = 1\), either \(\xi _1 \ne 0\) or \(\xi _2 \ne 0\). Assume \(\xi _1 \ne 0\). In view of \((\xi \vert \nu ) = 0\) and \(\vert \nu \vert = 1\), this implies \(\nu _2 \ne 0\). Thus, from (4.11), we obtain
Plugging this into (4.10) immediately yields the claim. The case \(\xi _2 \ne 0\) follows analogously. \(\square \)
We proceed by showing that Hibler’s operator subject to Dirichlet boundary conditions fulfills the Lopatinskii–Shapiro condition. For the formulation of the latter condition in the context of parabolic boundary value problems subject to general boundary conditions, see e.g., Denk et al. (2003, 2004, 2007).
In our context of Hibler’s operator subject to Dirichlet boundary conditions, the Lopatinskii–Shapiro condition reads as follows: For all \(x_0 \in \partial \Omega \), all \(\xi \in {\mathbb {R}}^2\) with \((\xi ,\nu (x_0))=0\), and all \(\lambda \in {\mathbb {C}}\) satisfying \({{\,\mathrm{Re}\,}}\lambda \ge 0\) and \(|\xi | + |\lambda | \ne 0\), any solution \(w \in C_0({\mathbb {R}}_+;{\mathbb {C}}^2)\) of the ordinary differential equation in \({\mathbb {R}}_+\)
equals zero.
Proposition 4.3
Let \(p,q \in (1,\infty )\) and \(\mu \in (\frac{1}{p},1]\) such that (2.16) holds. Then, for fixed \(v_0 \in V_\mu \), the principal part of Hibler’s operator \(\mathcal {A}^\mathrm{H}(v_0)\) subject to homogeneous Dirichlet boundary conditions satisfies the Lopatinskii–Shapiro condition.
Proof
Taking the inner product of the above equation with a solution w, we obtain
Integrating over \({\mathbb {R}}_{+}\) and integrating by parts yields
Our aim is to deduce from (4.13) that \(w \equiv 0\) for each solution \(w \in H_2^2 ({\mathbb {R}}_{+};{\mathbb {C}}^2)\) and thus for \(w \in C_b ^1({\mathbb {R}}_{+};{\mathbb {C}}^2)\).
Taking real parts in (4.13), we see by (4.7) that
Assuming \(\frac{\hbox {d}}{\hbox {d}y} \vert w(y) \vert ^2 = 0\) for all \(y > 0\) yields that \(\vert w(y) \vert \) is constant on \({\mathbb {R}}_{+}\) and that consequently \(w(y)=0\) on \({\mathbb {R}}_{+}\).
We calculate \({\frac{\text{ d }}{\text{ d } y} \vert w(y) \vert ^2 = 2 {{\, \mathrm {Re}\,}}\bigl (\frac{\hbox {d}}{\text{ d }y}w(y) \vert w(y)\bigr ) = - 2 {{\,\mathrm {Im}\,}}(D_y w(y) \vert w(y))}\). Suppose now that there exists \({y_0 > 0}\) such that \(\frac{\hbox {d}}{\hbox {d}y} \vert w(y) \vert ^2\) does not vanish at \(y_0\). Then, by smoothness of w, there exists a neighborhood \({U \subset {\mathbb {R}}_{+}}\) of \(y_0\) with \({\frac{\hbox {d}}{\hbox {d}y} \vert w(y) \vert ^2 \ne 0}\) for all \({y \in U}\). Then also \({{{\,\mathrm{Im}\,}}(D_y w(y) \vert w(y)) \ne 0}\) for all \({y \in U}\). Setting \({u:= D_y w(y) \in {\mathbb {C}}^2}\) and \({v:= w(y) \in {\mathbb {C}}^2}\) for \({y \in U}\) we see that \({{{\,\mathrm{Im}\,}}(u,v) \ne 0}\) for all \({y \in U}\) as well. Consequently, (4.8) and (4.7) yield
Combining this with relation (4.7) contradicts, however, condition (4.14). Thus, it holds that \(w \equiv 0\). \(\square \)
We recall that for \(1<r<\infty \), the \(L_r\)-realization \(A^\mathrm{H}_\mathrm{D}(v_0)\) of \(\mathcal {A}^\mathrm{H}(v_0)\) subject to Dirichlet boundary conditions is given by
We will now prove the maximal \(L_s\)-regularity property for \(A^\mathrm{H}_\mathrm{D}(v_0)\) in the \(L_r\)-setting, where \(1< s,r < \infty \).
Theorem 4.4
Let \(p,q,r,s \in (1,\infty )\) and \(\mu \in (\frac{1}{p},1]\) such that (2.16) holds and let \(v_0 \in V_\mu \) be fixed. Then, there exists \(\omega _0 \in {\mathbb {R}}\) such that for all \(\omega >\omega _0\)
(a) \(A^\mathrm{H}_\mathrm{D}(v_0) + \omega \) has the property of maximal \(L_s[0,\infty )\)-regularity on \(L_r(\Omega ;{\mathbb {R}}^2)\),
(b) \(A_\mathrm{D}^\mathrm{H}(v_0)+ \omega \) admits a bounded \(H^\infty \)-calculus on \(L_r(\Omega ;{\mathbb {R}}^2)\).
Proof
By Proposition 4.1, for fixed \(v_0 \in V_\mu \), the principal part of Hibler’s operator \(\mathcal {A}^\mathrm{H}(v_0)\) is a parameter–elliptic operator with continuous and bounded coefficients on \({\overline{\Omega }}\) having angle of ellipticity \(\phi _{\mathcal {A}^\mathrm{H}(v_0)}=0\). Furthermore, Proposition 4.3 tells us that the principal part of Hibler’s operator \(\mathcal {A}^\mathrm{H}(v_0)\) subject to homogeneous Dirichlet boundary conditions satisfies the Lopatinskii–Shapiro condition. Since the coefficients of the lower-order terms of \(\mathcal {A}^\mathrm{H}(v_0)\) are smooth, the first assertion follows from the results in Denk et al. (2003, 2007).
The second assertion follows by the results in Denk et al. (2004) provided the top-order coefficients of \(\mathcal {A}^\mathrm{H}_\mathrm{D}(v_0)\) are Hölder continuous. The latter condition is satisfied due to the embedding \(B_{qp}^{2\mu -2/p}(\Omega ) \hookrightarrow C^{1,\alpha }({\overline{\Omega }})\). \(\square \)
5 Functional Analytic Properties of Hibler’s Operator
Hibler’s operator enjoys many interesting properties, some of which we collect in the following.
Proposition 5.1
Let \({\Omega \subset {\mathbb {R}}^2}\) be a bounded domain with boundary of class \(C^2\), \({1< p,q,r < \infty }\), \({\mu \in (\frac{1}{p},1]}\) such that (2.16) is satisfied, and for \(v_0 \in V_\mu \), let Hibler’s operator \(A^\mathrm{H}_\mathrm{D}(v_0)\) on \(L_r(\Omega ;{\mathbb {R}}^2)\) with domain \(D(A_\mathrm{D}^\mathrm{H}(v_0))\) be defined as in (4.15). Then,
-
(a)
\( -A_\mathrm{D}^\mathrm{H}(v_0)\) generates an analytic semigroup \({e}^{-tA^\mathrm{H}_\mathrm{D}(v_0)}\) on \(L_r(\Omega ;{\mathbb {R}}^2)\),
-
(b)
\(A_\mathrm{D}^\mathrm{H}(v_0)\) is an operator with compact resolvent,
-
(c)
the spectrum \(\sigma (A^\mathrm{H}_\mathrm{D}(v_0))\) of \(A^\mathrm{H}_\mathrm{D}(v_0)\) viewed as an operator on \(L_r(\Omega )^2\) is r-independent,
-
(d)
for \(\alpha \in (0,1)\) and \(\omega \) as in Theorem 4.4
$$\begin{aligned} D((A^\mathrm{H}_\mathrm{D}(v_0)+\omega )^\alpha ) \simeq [L_r(\Omega ;{\mathbb {R}}^2),D(A_\mathrm{D}^\mathrm{H}(v_0))]_{\alpha } = {\left\{ \begin{array}{ll} \{u \in H^{2\alpha }_r(\Omega ): u|_{\partial \Omega } =0\}, \alpha \in (1/2r,1], \\ H^{2\alpha }_r(\Omega ): \alpha \in [0,1/2r), \end{array}\right. }\end{aligned}$$ -
(e)
The Riesz transform \(\nabla (A^\mathrm{H}_\mathrm{D}(v_0)+\omega )^{-1/2}\) of \(A^\mathrm{H}_\mathrm{D}(v_0)\) is bounded on \(L_r(\Omega )^2\).
Proof
Assertions (a) follows by standard arguments. The compact embedding \(D(A^\mathrm {H}_\mathrm {D}(v_0)) \hookrightarrow L_r(\Omega )^2\) implies that \(A^\mathrm{H}_\mathrm{D}(v_0)\) has compact resolvent and that thus \(\sigma (A^\mathrm{H}_\mathrm{D}(v_0))\) is independent of \(r \in (1,\infty )\). Assertion (d) follows from the fact that \(A^\mathrm{H}_\mathrm{D}(v_0)+\omega \) admits a bounded \(H^\infty \)-calculus on \(L_r(\Omega )^2\) by Theorem 4.4. Finally, assertion (e) is obtained by noting that \(D((A^\mathrm{H}_\mathrm{D}(v_0)+\omega )^{1/2}) \subset H^1_r(\Omega )\). \(\square \)
6 Proof of Theorem 2.1
We recall from Sect. 2 that the ground space \(X_0\) for \(q \in (1,\infty )\) is given by
The regularity space \(X_1\) is defined as
Since we are considering solutions within the class
where \(J=(0,T)\) with \(0<T\le \infty \) is an interval and \(\mu \in (1/p,1]\) indicates a time weight, the time trace space of this class is given by
provided \(p \in (1,\infty )\) and \(\mu \in (1/p,1]\). Note that
provided (2.16) is satisfied.
For \(\omega \) as in Theorem 4.4, we now consider the operator \(A_\omega (v_0)\) on \(X_0\) with domain \(X_1\) given by the upper triangular matrix
as well as
where F is given as in (2.13).
Lemma 6.1
Let \(p,q \in (1,\infty )\), \(\mu \in (1/p,1]\) such that (2.16) is satisfied and assume that \(v_0=(u_0,h_0,a_0) \in V_\mu \). Let \(J = [0,T)\) for some \(0< T < \infty \). Then, \(A_\omega (v_0)\) has maximal \(L_p(J)\)-regularity on \(X_0\).
Proof
By assumption, we have \(h_0 \ge \kappa \) for some \(\kappa >0\) and (2.16) implies that \(1/h_0 \in C^1({\overline{\Omega }})\). Since \(\Delta _\mathrm{N}\) as well as \(A^\mathrm{H}_\mathrm{D}(v_0)+\omega \) and \(\frac{1}{h_0}(A^\mathrm{H}_\mathrm{D}(v_0)+\omega )\) have the maximal \(L_p(J)\)-regularity property on \(X_0\) by Theorem 4.4, the upper triangular structure of \(A_\omega (v_0)\) implies that also \(A_\omega (v_0)\) has the maximal \(L_p(J)\)-regularity property on \(X_0\).
We now show that \((A_\omega ,F_\omega ) \in C^{1-}(V_\mu ;{{\mathcal {L}}}(X_1,X_0) \times X_0)\) for \(\mu \in (1/p,1]\) satisfying (2.16) and where A and F are defined as in (6.2) and (6.3). Recall that \(V_\mu \) is an open subset of \(X_{\gamma ,\mu }\) such that all \((u,h,a) \in V_\mu \) satisfy \(h \ge \kappa \) for some \(\kappa >0\).
Lemma 6.2
Let \(p,q \in (1,\infty )\), \(\mu \in (1/p,1]\) such that (2.16) is satisfied. Suppose that \(A_\omega \) and \(F_\omega \) are defined as in (6.2) and (6.3) and let \(v_0=(u_0,h_0,a_0) \in V_\mu \). Then, there exists \(r_0>0\) and a constant \(L>0\) such that \({\overline{B}}_{X_{\gamma ,\mu }}(v_0,r_0) \subset V_\mu \) and
for all \(v_1,v_2 \in {\overline{B}}_{X_{\gamma ,\mu }}(v_0,r_0)\) and all \(w \in X_1\).
Proof
Choose \(r_0>0\) small enough such that \(v_1, v_2 \in {\overline{B}}_{X_{\gamma ,\mu }}(v_0,r_0) \subset V_\mu \). For \(w=(u,h,a) \in X_1\), we then obtain
To prove the assertion for \(F_\omega \), we start with the convective term \(u \nabla u\). Hölder’s inequality and the embedding \(X_{\gamma ,\mu } \hookrightarrow L_s \cap H_s ^1\) for \(s = qr\) and \(s=qr'\) imply
A similar argument shows that
Furthermore, note that \(\tau _{\mathrm{{atm}}}\) is constant in v, and thus \({\frac{\tau _{\mathrm{{atm}}}}{\rho _{\mathrm{{ice}}}h_0}}\) is Lipschitz continuous in v. Concerning \(\tau _{\mathrm{{ocean}}}\), we may assume that \(U_{\mathrm{{atm}}}= 0\) (otherwise consider \(u + U_{\mathrm{{atm}}}\)). It thus suffices to show that \( v \mapsto \frac{1}{h}{u \vert u \vert }\) is Lipschitz continuous viewed as a mapping from \(V_\mu \) to \(L_q\). The term \(\frac{1}{\rho _{\mathrm{{ice}}}h}\omega u\) is treated in the same way.
Finally, we consider the terms \(S_h\) and \(S_a\) defined as in (2.8) and (2.9), respectively. By assumption, \(f \in C^1\) and hence \(S_h\) as well as \(S_a\) are Lipschitz continuous in v. \(\square \)
The assertion of Theorem 2.1 follows hence by the local existence theorem for quasilinear evolution equation as described in Thm. 5.1.1 in Prüss and Simonett (2016).
7 Proof of Theorem 2.3
Throughout this section, we consider \(p,q \in (1,\infty )\) and \(\mu \in (\frac{1}{p},1]\) such that (2.16) holds. Moreover, we abbreviate \(P(h_*,a_*)\) by \(P_*\), i.e.,
We study equilibria in the case that no external forces are present in the momentum equation, i.e.,
and neglect external freezing and melting effects by setting
For A as in (2.12) and the simplified semilinear right-hand side \(F_s\) given by
we prove similarly as in Sect. 6 that there is an open set \(V \subset V_\mu \subset X_{\gamma ,\mu }\) such that
We denote by \({{\mathcal {E}}}\subset V \cap X_1\) the set of equilibrium solutions of
An equilibrium solution \(v \in {{\mathcal {E}}}\) is characterized by \(v \in V \cap X_1\) and \(A(v) v = F_s(v)\).
For \(h_*\ge \kappa \) and \(a_*\ge 0\) constant in time and space, \(v_*= (0, h_*, a_*) \in V \cap X_1\) is an equilibrium solution of (7.2) due to \(A(v_*)v_*= 0 = F_s(v_*)\).
To prove Theorem 2.3, we aim to apply the generalized principle of linearized stability, see Prüss et al. (2009) or Prüss and Simonett (2016). Note that we already verified that \(v_*\in V \cap X_1\) is an equilibrium of (7.2) and that \((A,F_s)\) satisfy (7.1). Consider next the linearization of (7.2) at \(v_*\) which reads as
Computing \(A_0 v\), we see first that \(A(v_*)v\) is given by
where
Secondly, we deduce that \((A^\prime (v_*)v)v_*= 0\) for all \(v \in X_1\) and that
The linearization \(A_0\) hence becomes
Lemma 7.1
If \(v_*\) is as above, then there exists \(\delta _*>0\) such that \(\sigma (A_0)\setminus \{0\} \subset {\mathbb {C}}_+\) holds for all \(0<\delta < \delta _*\). Furthermore, 0 is a semi-simple eigenvalue of \(A_0\) and \(N(A_0)\) has dimension 2.
Proof
To locate the spectrum of \(A_0\), we test the equation \((\lambda + A_0)v = 0\) by \(v=(u,h,a)\) and use integration by the vector parts, which leads to
Thanks to the Dirichlet boundary condition for u and employing (4.4) as well as Korn’s and Poincaré’s inequality, we deduce that
for some constant \(C_*> 0\) independent of \(\delta \) and u. Now, the remaining terms in (7.3) can be absorbed: First determine \(\gamma _h,\gamma _a > 0\) depending in particular on \(h_*, a_*, d_h\) and \(d_a\) such that
and, similarly, such that
Then, choose \(\delta _*> 0\) sufficiently small to ensure that \(\gamma _h + \gamma _a < \frac{C_*}{\sqrt{\delta _*}}\). In particular, this implies that for all \(\delta < \delta _*\), there exists \(C_\delta > 0\) such that
The relation in (7.5) can only hold provided that \(\lambda \) is real and that \(\lambda \le 0\). Hence, \(\sigma (A_0)\setminus \{0\} \subset {\mathbb {C}}_+\). For \(\lambda = 0\), we infer that \(u=0\) and h as well as a are constant. This implies that 0 is a semi-simple eigenvalue of \(A_0\) and that \(N(A_0)\) has dimension 2. \(\square \)
Lemma 7.2
Near \(v_*\), the set of equilibria \({{\mathcal {E}}}\) is a \(C^1\)-manifold in \(X_1\), and the tangent space of \({{\mathcal {E}}}\) at \(v_*\) is isomorphic to \(N(A_0)\).
Proof
Consider equilibria \(v \in V \cap X_1\) such that \(\Vert v - v_*\Vert _{X_{\gamma ,\mu }} < r\) for given \(r > 0\). The resulting equation for such v is
We set the constants
and test the above equation with \((u, C_h h, C_a a)\) to obtain
Using the symmetry of \({\mathbb {S}}\), the estimate \(P(v)\ge P_*\kappa \exp (-c(1-\alpha ))\), the estimate \(\triangle ^2_\delta (\varepsilon ) \le C_e r\) and Korn’s and Poincare’s inequalities, the first term on the right-hand side satisfies
for some constant \(C_V >0\) independent of \(\delta ,r\) and v. We show how terms without sign in (7.7) can now be absorbed. We first discuss the case \(a_*\ne 0\) and remark on the case \(a_*= 0\) below. First note that using \(\Vert v - v_*\Vert _{X_{\gamma ,\mu }} < r\) and any bound on \(r>0\),
for a suitable constant \(C_*> 0\) that is independent of \(\delta , r\) and v. Secondly, we calculate
and use that part of this expression cancels with the terms
in (7.7). It remains to check that due to the particular choice of \(C_h,C_a\) in (7.6), we find that for a (possibly increased) constant \(C_*> 0\),
and similarly
and hence the terms
and
are controlled.
In summary, inserting the above estimates into Eq. (7.7), we conclude that
Hence, if \(r > 0\) is sufficiently small, then (7.7) implies
This shows that for \(v=(u,h,a) \in V_*\) with \(\Vert v - v_*\Vert _{X_{\gamma ,\mu }} < r\), we have \(u=0\) and h as well as a must be constant. In particular, \({{\mathcal {E}}}= N(A_0)\) is valid in a neighborhood of \(v_*\).
The case \(a_*= 0\) can be included by a slight adjustment of the argument. Replace (7.8) by
and directly estimate
as well as
to conclude as before. \(\square \)
Lemma 7.3
For \(v_*\) as above, \(A(v_*)\) has the property of maximal \(L_s\)-regularity on \(L_r(\Omega ;{\mathbb {R}}^2)\).
Proof
We know from Theorem 4.4 that there is \(\omega _0 \in {\mathbb {R}}\) such that \(A^\mathrm{H}_\mathrm{D}(v_*) + \omega \) has the maximal \(L_s\)-regularity on \(L_r(\Omega ;{\mathbb {R}}^2)\) for all \(\omega > \omega _0\). Considering the eigenvalue equation for \(A^\mathrm{H}_\mathrm{D}(v_*)\) it follows by (7.4) that
Thus, \(s(-A^\mathrm{H}_{D,2}(v_*)) < 0\) and \(A^\mathrm{H}_{D,2}(v_*)\) is invertible in \(L_2(\Omega )^2\). Due to compact embeddings, \(A^\mathrm{H}_\mathrm{D}(v_*)\) has compact resolvent and hence the spectrum of \(A^\mathrm{H}_\mathrm{D}(v_0)\) is r-independent, and we see that \(s(-A^\mathrm{H}_{D,2}(v_*))= s(-A^\mathrm{H}_\mathrm{D}(v_*))<0\). It can be shown that \(\omega _0\) can be chosen to be equal to the spectral bound \(s(-A^\mathrm{H}_\mathrm{D}(v_*))\) of \(A^\mathrm{H}_\mathrm{D}(v_*)\), i.e., \(\omega _0=s(-A^\mathrm{H}_\mathrm{D}(v_*))\), which implies that \(A^\mathrm{H}_\mathrm{D}(v_*)\) has the maximal \(L_s\)-regularity on \(L_r(\Omega ;{\mathbb {R}}^2)\). The triangular structure of \(A(v_*)\) implies that this the latter property holds also for \(A(v_*)\). \(\square \)
Summarizing we see that Lemmas 7.1, 7.2 and 7.3 imply that the assumptions of the principle of linearized stability described as in Prüss et al. (2009) or Prüss and Simonett (2016) are fulfilled. The assertion of Theorem 2.3 follows thus by this principle.
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Brandt, F., Disser, K., Haller-Dintelmann, R. et al. Rigorous Analysis and Dynamics of Hibler’s Sea Ice Model. J Nonlinear Sci 32, 50 (2022). https://doi.org/10.1007/s00332-022-09805-w
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DOI: https://doi.org/10.1007/s00332-022-09805-w
Keywords
- Hibler’s sea ice model
- Local and global well-posedness
- Viscous–plastic stress tensor
- Stability of equilibria