A variational approach to hyperbolic evolutions and fluid-structure interactions

Abstract

We introduce a two time-scale scheme which allows to extend the method of minimizing movements to hyperbolic problems. This method is used to show the existence of weak solutions to a fluid-structure interaction problem between a nonlinear, visco-elastic, ndimensional bulk solid governed by a hyperbolic evolution and an incompressible fluid governed by the (n-dimensional) Navier-Stokes equations for n ≥ 2. Contents 1. Introduction 1 1.1. Setup 3 1.2. Main result 5 1.3. Mechanical and analytical restrictions on the energy/dissipation functional 9 1.4. Outlook and further applicability 11 1.5. Notation 12 2. Minimizing movements for fluid-structure interactions. 13 2.1. Preliminary analysis 14 2.2. Proof of Theorem 2.2 16 2.3. The example Energy-Dissipation pair 26 3. Minimizing movements for hyperbolic evolutions 28 3.1. The time-delayed problem 30 3.2. Proof of Theorem 3.2 33 3.3. Remarks 37 4. The unsteady fluid-structure interaction problem 39 4.1. An intermediate, time delayed model 40 4.2. Proof of Theorem 1.2 46 4.3. Remarks on the full problem 56 Appendix A. 57 A.1. Some technical lemmata 57 A.2. Proof of Proposition 2.22 59 Acknowledgments 62 References 62

Rational Decay of A Multilayered Structure-Fluid PDE System

“…Examples of single layered FSI -i.e., only one elastic PDE (describing three dimensional bulk elasticity or some lower-dimensional model of plate/shell type) models the structural dynamics; the displacement along the interaction interface is not modeled via any elastic equation-appear extensively in the literature,see e.g. [9,11,13,21,22,27,32,38] and references within. However, many biomedical devices (such as stents) are being developed with the view that vascular wall structures are composed of composite materials and not of single layer; see [19,20,24,25,39].…”

“…In view of relation (13), we should strive to majorize solution Φ of (10) in norm in terms of the static heat dissipation. With this theme in mind, from the mechanical compatibility conditions in (5), and the resolvent relations and matching velocity BC's in (11), we have for j = 1, ..., K,…”

Existence of a weak solution to a nonlinear fluid-structure interaction problem with heat exchange

“…The results on the existence of a weak solution typically deal with FSI problems where the elastic structure is described by a lower dimensional model of a plate/shell type, see [47,8,21,37,46,55] and references within. Exceptions are works [1,48] where the existence of a weak solution to FSI problems involving regularized, nonlinear, 3D viscoelastic structure and linear multilayered structure, respectively, were proven. All these works consider large data case and a solution existing as long as geometry does not degenerate, i.e.…”

“…Remark 2.1. (1) The derivation of coupled momentum equation for smooth solutions is standard and we refer to [5,56] for more details. However, the coupled entropy inequality appears here for the first time and it is derived in Appendix B.…”

Construction of a right inverse for the divergence in non-cylindrical time dependent domains

“…Moreover, it can be used for constructing approximating sequences of testfunctions, which is usually a very delicate technical issue since the space of test functions depends on the motion of the solid in that setting. Up to the authors' knowledge, so far pressure reconstruction as well as the respective solenoidal approximation results are only available in smooth space-time domains [8].…”