Equations of second order in time with quasilinear damping: existence in Orlicz spaces via convergence of a full discretisation

Emmrich, Etienne; Šiška, David; Wróblewska-Kamińska, Aneta

doi:10.1002/mma.3706

Cited by 4 publications

(3 citation statements)

References 24 publications

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“…Assuming that A, B are operators on ``nice"" function spaces and by considering a sequence of approximating problems based on temporal discretization, the authors prove the existence of a weak solution to this doubly nonlinear problem. We also mention the related work [17], where the authors consider…”

Section: 2)mentioning

confidence: 99%

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

Bulíček¹,

Patel²,

Şengül³

et al. 2022

SIAM J. Math. Anal.

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We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The constitutive relation, involving the Cauchy stress, the small strain tensor, and the symmetric velocity gradient, is given in an implicit form. For a large class of these implicit constitutive relations, we establish the existence and uniqueness of a global-in-time large-data weak solution. Then we focus on the class of socalled limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises. The Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfill natural compatibility conditions.

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Section: 2)mentioning

confidence: 99%

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

Bulíček¹,

Patel²,

Şengül³

et al. 2022

SIAM J. Math. Anal.

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“…For the last step of the proof, it will be crucial to use the limit equation (4.5) not only for test functions in V ⊗ C 1 ([0, T ]), but for a more general class of test functions. We will use the following approximation result almost identical to [10,Lemma 4.3].…”

Section: Existence Via Convergence Of Approximate Solutionsmentioning

confidence: 99%

“…The remaining step is to show that α = σ(∇u). To this end, we use a variant of Minty's trick adapted to the case of nonreflexive Orlicz spaces (see also [10,11,19,27]). For k ≥ 0 we define…”

Section: Identification Of Initial and Final Valuesmentioning

confidence: 99%

Convergence of a full discretization for a second-order nonlinear elastodynamic equation in isotropic and anisotropic Orlicz spaces

Ruf

2017

Z. Angew. Math. Phys.

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In this paper, we study a second-order, nonlinear evolution equation with damping arising in elastodynamics. The nonlinear term is monotone and possesses a convex potential but exhibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation. Moreover, we show uniqueness in a class of sufficiently smooth solutions and provide an a priori error estimate for the temporal semidiscretization.Mathematics Subject Classification (2010). 35L20; 47J35; 47H05; 65M12; 65M06; 35M60.for ε > 0, as well as for ε = 0 and presents numerical experiments [28]. The limit case ε = 0 constitutes the elastodynamic equationfor which one cannot expect smooth solutions even for smooth initial data (see [2,26]). For an excellent survey of the literature concerning equation (1.3) see [8]. The remainder of this paper is structured as follows: In Section 2, we introduce the necessary notation, give a brief introduction to Orlicz spaces and compare the growth condition (1.2) with the restrictive ∆ 2 -condition. The description of the numerical method we employ, the construction of the Galerkin scheme, the proof of existence and uniqueness of the numerical solution, and the derivation of a priori estimates for the fully discrete solution and the discrete time derivative follow in Section 3. Finally, in Section 4 we show convergence towards and, thus, existence of an exact solution, as well as its uniqueness (under additional regularity assumptions) and an error estimate for the temporal semidiscretization. The appendix contains an elementary lemma concerning the separability of the space for wich we want to construct the Galerkin scheme.

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Well-posedness of a fully nonlinear evolution inclusion of second order

Bacho

2024

J Elliptic Parabol Equ

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Equations of second order in time with quasilinear damping: existence in Orlicz spaces via convergence of a full discretisation

Cited by 4 publications

References 24 publications

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

Existence and Uniqueness of Global Weak Solutions to Strain-Limiting Viscoelasticity with Dirichlet Boundary Data

Convergence of a full discretization for a second-order nonlinear elastodynamic equation in isotropic and anisotropic Orlicz spaces

Well-posedness of a fully nonlinear evolution inclusion of second order

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