We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid-solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory.

We consider finite energy solutions of a wave equation with supercritical nonlinear sources and nonlinear damping. A distinct feature of the model under consideration is the presence of nonlinear sources on the boundary driven by Neumann boundary conditions. Since Lopatinski condition fails to hold (unless the dim(Ω) = 1), the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves the strong interaction between the source and the damping. Thus, it is not surprising that existence theory for this class of problems has been established only recently. However, the uniqueness of weak solutions was declared an open problem. The main result in this work is uniqueness of weak solutions. This result is proved for the same (even larger) class of data for which existence theory holds. In addition, we prove that weak solutions are continuously depending on initial data and that the flow corresponding to weak and global solutions is a dynamical system on the finite energy space.

We consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources);(2) show that the corresponding solutions satisfy Hadamard wellposedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping.

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This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well-posed in the finite-energy space H 1 (Ω) × L2 (∂Ω) with boundary data in L2 due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non-dissipative and are not locally Lipschitz operators from H 1 (Ω) into L2 (Ω), or L2 (∂Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.

We focus on the blow-up in finite time of weak solutions to the wave equation with interior and boundary nonlinear sources and dissipations. Our central interest is the relationship of the sources and damping terms to the behavior of solutions. We prove that under specific conditions relating the sources and the dissipations (namely p > m and k > m), weak solutions blow up in finite time.

We study the well-posedness of a total linearization, with respect to a perturbation of the external forcing, of a free-boundary nonlinear elasticity-incompressible fluid interaction. The total linearization for the coupling modeled by the Navier-Stokes equations and the nonlinear equations of elastodynamics was obtained recently in [L. Bociu and J.-P. Zolésio, Evol. Equ. Control Theory, 2 (2013), pp. 55-79]. The equations and the free boundary were linearized together, and the result turned out to be quite different from the usual coupling of classical linear models. New additional terms are present on the common interface, some of them involving boundary curvatures and boundary acceleration. These terms play an important role in the final linearized system and cannot be neglected; their presence also introduces new challenges in the well-posedness analysis, which proceeds to establish that the evolution operator associated to the linearized system can be represented as a bounded perturbation of a maximal dissipative semigroup generator. Introduction.A typical fluid-structure interaction system is composed of a solid either surrounding a fluid flow (arteries, pipelines, reservoir tanks, etc.) or surrounded by a fluid or gas (blood cells, marine animals, bridge supports, aircraft, automobiles, etc.). This model falls in the category of free boundary problems, since the motion of the common boundary between the solid and the fluid is one of the unknowns in the coupled system. The ubiquity of this interaction type has led to a rapidly growing research interest in this model. Its mathematical solvability [5,9,15,17,22,21,26,32,33,34,37,38,49,55,52], numerical approximations [23,29,39,54,53,44,46], and stability [14,27,28,31] have been intensively studied.Mathematically, this interaction is described by a partial differential equation (PDE) system that couples the parabolic (fluid) and hyperbolic (elasticity) phases, where the key issue is that the traces of the elastic component at the energy level are not defined via the standard trace theory. The loss of regularity induced by the hyperbolic component left the basic question of existence open until recently.The first major development appeared in [16], where the authors established local existence and regularity of solutions for the coupling of Navier-Stokes equations and a linear elasticity model, describing a body submerged in a fluid, under the assumptions of smooth initial data; specifically, the initial fluid velocity w 0 has Sobolev regularity H 5 , and the initial data for the elastic component (ϕ 0 , ϕ 1 ) belongs to H 3 × H 2 . Moreover, due to the incompressibility condition on the fluid, the uniqueness result

In this article we consider poro-elastic and poro-visco-elastic models inspired by problems in medicine and biology [12], and we perform sensitivity analysis on the solutions of these fluid-solid mixtures problems with respect to the imposed boundary data, which are the main drivers of the system. Moreover, we compare the results obtained in the elastic case vs. viscoelastic case, as it is known that structural viscosity of biological tissues decreases with age and disease. Sensitivity analysis is the first step towards optimization and control problems associated with these models, which is our ultimate goal.