Higher Regularity of a Coupled Parabolic-Hyperbolic Fluid-Structure Interactive System

2008 ◽  
Vol 15 (3) ◽  
pp. 403-437 ◽  
Author(s):  
George Avalos ◽  
Irena Lasiecka ◽  
Roberto Triggiani
Keyword(s):  
Fluid Velocity ◽  
Coupled System ◽  
Finite Energy ◽  
Elastic Structure ◽  
Regularity Of Solutions ◽  
Interactive System ◽  
Well Posedness ◽  
Mechanical Displacement ◽  
Fluid Coupling ◽  
Higher Regularity

Abstract This paper considers an established model of a parabolic-hyperbolic coupled system of two PDEs, which arises when an elastic structure is immersed in a fluid. Coupling occurs at the interface between the two media. Semigroup well-posedness on the space of finite energy for {𝑤, 𝑤𝑡, 𝑢} was established in [Contemp. Math. 440: 15–54, 2007]. Here, [𝑤, 𝑤𝑡] are the displacement and the velocity of the structure, while 𝑢 is the velocity of the fluid. The domain D(A) of the generator A does not carry any smoothing in the 𝑤-variable (its resolvent 𝑅(λ, A) is not compact on this component space). This raises the issue of higher regularity of solutions. This paper then shows that the mechanical displacement, fluid velocity, and pressure terms do enjoy a greater regularity if, in addition to the I.C. {𝑤0, 𝑤1, 𝑢0} ∈ D(A), one also has 𝑤0 in (𝐻2(Ω𝑠))𝑑.

scholarly journals Asymptotic behavior of nonlocal partial differential equations with long time memory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jiaohui Xu ◽  
Tomás Caraballo ◽  
José Valero
Keyword(s):  
Original Problem ◽  
Global Attractors ◽  
Standard Theory ◽  
Regularity Of Solutions ◽  
Well Posedness ◽  
Partial Differential ◽  
Long Time ◽  
Higher Regularity ◽  
With Memory ◽  
Autonomous Dynamical System

<p style='text-indent:20px;'>In this paper, it is first addressed the well-posedness of weak solutions to a nonlocal partial differential equation with long time memory, which is carried out by exploiting the nowadays well-known technique used by Dafermos in the early 70's. Thanks to this Dafermos transformation, the original problem with memory is transformed into a non-delay one for which the standard theory of autonomous dynamical system can be applied. Thus, some results about the existence of global attractors for the transformed problem are {proved}. Particularly, when the initial values have higher regularity, the solutions of both problems (the original and the transformed ones) are equivalent. Nevertheless, the equivalence of global attractors for both problems is still unsolved due to the lack of enough regularity of solutions in the transformed problem. It is therefore proved the existence of global attractors of the transformed problem. Eventually, it is highlighted how to proceed to obtain meaningful results about the original problem, without performing any transformation, but working directly with the original delay problem.</p>

scholarly journals Local Existence of Strong Solutions of a Fluid–Structure Interaction Model

2020 ◽  
Vol 22 (4) ◽  
Author(s):  
Sourav Mitra
Keyword(s):  
Stokes Equations ◽  
Local Existence ◽  
Interaction Model ◽  
Strong Solutions ◽  
Coupled System ◽  
Elastic Structure ◽  
Navier Stokes ◽  
Beam Equation ◽  
Navier Stokes Equations ◽  
System Coupling

AbstractWe are interested in studying a system coupling the compressible Navier–Stokes equations with an elastic structure located at the boundary of the fluid domain. Initially the fluid domain is rectangular and the beam is located on the upper side of the rectangle. The elastic structure is modeled by an Euler–Bernoulli damped beam equation. We prove the local in time existence of strong solutions for that coupled system.

scholarly journals A semidiscrete Galerkin scheme for a coupled two-scale elliptic–parabolic system: well-posedness and convergence approximation rates

2020 ◽  
Vol 60 (4) ◽  
pp. 999-1031
Author(s):  
Martin Lind ◽  
Adrian Muntean ◽  
Omar Richardson
Keyword(s):  
Parabolic Equations ◽  
Spatial Scales ◽  
A Priori ◽  
Coupled System ◽  
Scale Model ◽  
Reactive Flow ◽  
Well Posedness ◽  
Approximation Rates ◽  
Priori Error Estimates ◽  
Model Equations

AbstractIn this paper, we study the numerical approximation of a coupled system of elliptic–parabolic equations posed on two separated spatial scales. The model equations describe the interplay between macroscopic and microscopic pressures in an unsaturated heterogeneous medium with distributed microstructures as they often arise in modeling reactive flow in cementitious-based materials. Besides ensuring the well-posedness of our two-scale model, we design two-scale convergent numerical approximations and prove a priori error estimates for the semidiscrete case. We complement our analysis with simulation results illustrating the expected behaviour of the system.

scholarly journals Well-posedness and regularity results for a dynamic Von Kármán plate

1995 ◽  
Vol 18 (2) ◽  
pp. 237-244
Author(s):  
M. E. Bradley
Keyword(s):  
Strong Solutions ◽  
Free Edge ◽  
Regularity Of Solutions ◽  
Well Posedness ◽  
Boundary Damping ◽  
Von Karman ◽  
Edge Conditions ◽  
Von Kármán Plate ◽  
Regularity Results ◽  
Von Kármán

We consider the problem of well-posedness and regularity of solutions for a dynamic von Kármán plate which is clamped along one portion of the boundary and which experiences boundary damping through “free edge” conditions on the remainder of the boundary. We prove the existence of unique strong solutions for this system

A model for the two-way propagation of water waves in a channel

1976 ◽  
Vol 79 (1) ◽  
pp. 167-182 ◽  
Author(s):  
Jerry L. Bona ◽  
Ronald Smith
Keyword(s):  
Water Waves ◽  
Open Channel ◽  
Boussinesq Equations ◽  
Coupled System ◽  
Finite Amplitude ◽  
Regularity Of Solutions ◽  
Initial Value ◽  
One Dimensional ◽  
Image Position ◽  
Continuous Dependence Of Solutions

Global existence, uniqueness and regularity of solutions and continuous dependence of solutions on varied initial data are established for the initial-value problem for the coupled system of equationsThis system has the same formal justification as a model for the two-way propagation of (one-dimensional) long waves of small but finite amplitude in an open channel of water of constant depth as other versions of the Boussinesq equations. A feature of the analysis is that bounds on the wave amplitude η are obtained which are valid for all time.

General decay in a Timoshenko-type system with thermoelasticity with second sound

2015 ◽  
Vol 4 (4) ◽  
pp. 263-284 ◽  
Author(s):  
Mohamed Ali Ayadi ◽  
Ahmed Bchatnia ◽  
Makram Hamouda ◽  
Salim Messaoudi
Keyword(s):  
Group Theory ◽  
Type System ◽  
General Decay ◽  
Regularity Of Solutions ◽  
Well Posedness ◽  
Second Sound ◽  
Semi Group ◽  
Stability Number ◽  
Timoshenko System ◽  
Relaxation Functions

AbstractIn this article, we consider a vibrating nonlinear Timoshenko system with thermoelasticity with second sound. We discuss the well-posedness and the regularity of solutions using the semi-group theory. Moreover, we establish an explicit and general decay result for a wide class of relaxation functions, which depend on a stability number μ.

scholarly journals Global well-posedness of the Chern-Simons-Higgs equations with finite energy

2013 ◽  
Vol 33 (6) ◽  
pp. 2531-2546 ◽  
Author(s):  
Sigmund Selberg ◽  
◽  
Achenef Tesfahun
Keyword(s):  
Finite Energy ◽  
Well Posedness ◽  
Chern Simons
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