Daniel Toundykov

Effects of Multiscale Anisotropy on Basin and Hyporheic Groundwater Flow

Various subsurface flow systems exhibit a combination of small-scale to large-scale anisotropy in hydraulic conductivity (K). The large-scale anisotropy results from systematic trends (e.g., exponential decrease or increase) of K with depth. We present a general two-dimensional solution for calculation of topography-driven groundwater flow considering both small- and large-scale anisotropy in K. This solution can be applied to diverse systems with arbitrary head distribution and geometry of the water table boundary, such as basin or hyporheic flow. In a special case, this solution reduces to the well-known Tóth model of uniform isotropic basin. We introduce an integral measure of flushing intensity that quantifies flushing at different depths. Using this solution, we simulate heads and streamlines and provide analyses of flow structure in the flow domain, relevant to basin analyses or hyporheic flow. It is shown that interactions between small-scale anisotropy and large-scale anisotropy strongly control the flow structure. In the classic Tóth flow model, the flushing intensity curves exhibit quasi-exponential decrease with depth. The new measure is capable of capturing subtle changes in the flow structure. Our study shows that both small- and large-scale anisotropy characteristics have substantial effects that need to be integrated into analysis of topography-driven flow.

Geometrically constrained stabilization of wave equations with Wentzell boundary conditions

Uniform stabilization of wave equation subject to second-order boundary conditions is considered in this article. Both dynamic (Wentzell) and static (with higher derivatives in space only) boundary conditions are discussed. In contrast to the classical wave equation where stabilization can be achieved by applying boundary velocity feedback, for a Wentzell-type problem boundary damping alone does not cause the energy to decay uniformly to zero. This is the case for both dynamic and static second-order conditions. In order to achieve uniform decay rates of the associated energy, it is necessary to dissipate part of the collar near the boundary. It will be shown how a combination of partially localized boundary feedback and partially localized collar feedback leads to uniform decay rates that are described by a nonlinear differential equation. This goal is attained by combining techniques used for stabilization of ‘unobserved’ Neumann conditions with differential geometry techniques effective for stabilization on compact manifolds. These lead to a construction of special non-radial multipliers which are geometry dependent and allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.

Finite-dimensional attractor for a composite system of wave/plate equations with localized damping

The long-term behaviour of solutions to a model for acoustic–structure interactions is addressed; the system consists of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are the existence of a global attractor for the dynamics generated by this composite system as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension—established in a previous work by Bucci et al (2007 Commun. Pure Appl. Anal. 6 113–40) only in the presence of full-interior acoustic damping—holds even in the case of localized dissipation. This nontrivial generalization is inspired by, and consistent with, the recent advances in the study of wave equations with nonlinear localized damping.

Wave equations with super-critical interior and boundary nonlinearities

Abstract: This article presents a unified overview of the latest, to date, results on boundary value problems for wave equations with super-critical nonlinear sources on both the interior and the boundary of a bounded domain @W@?R^n. The presented theorems include Hadamard local wellposedness, global existence, blow-up and non-existence theorems, as well as estimates on the uniform energy dissipation rates for the appropriate classes of solutions.

Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources

This paper investigates a quasilinear wave equation with Kelvin-Voigt damping, utt − Δpu − Δut = f(u), in a bounded domain Ω ⊂ ℝ3 and subject to Dirichlet boundary conditions. The operator Δp, 2 < p < 3, denotes the classical p-Laplacian. The nonlinear term f(u) is a source feedback that is allowed to have a supercritical exponent, in the sense that the associated Nemytskii operator is not locally Lipschitz from W01,p(Ω) into L2(Ω). Under suitable assumptions on the parameters, we prove existence of local weak solutions, which can be extended globally provided the damping term dominates the source in an appropriate sense. Moreover, a blow-up result is proved for solutions with negative initial total energy.

An analytical approach for flow analysis in aquifers with spatially varying top boundary.

Analytical models of groundwater flow with a spatially varying elevation of a top boundary are widely used. However, a vast majority of previous analytical studies truncated the irregularly shaped top section with little to no analyses of the shortcomings of the approximate solutions for the resulting rectangles or parallelepipeds. We present an analytical approach based on a perturbation technique that treats complete domains. It is especially accurate near the top boundary, where fluid circulation is most pronounced and higher accuracy is typically needed, such as in regional or hyporheic systems flow. The approach is illustrated by analyzing flow for a Tóthian unit basin.

Well-Posedness Analysis for a Linearization of a Fluid-Elasticity Interaction

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. Zolesio, 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 max...

A Nonlocal Biharmonic Operator and its Connection with the Classical Analogue

We consider a singular integral operator as a natural generalization to the biharmonic operator that arises in thin plate theory. The operator is built in the nonlocal calculus framework defined in (Math Models Methods Appl Sci 23(03):493–540, 2013) and connects with the recent theory of peridynamics. This framework enables us to consider non-smooth approximations to fourth-order elliptic boundary-value problems. For these systems we introduce nonlocal formulations of the clamped and hinged boundary conditions that are well-defined even for irregular domains. We demonstrate the existence and uniqueness of solutions to these nonlocal problems and demonstrate their L2-strong convergence to functions in W2,2 as the nonlocal interaction horizon goes to zero. For regular domains we identify these limits as the weak solutions of the corresponding classical elliptic boundary-value problems. As a part of our proof we also establish that the nonlocal Laplacian of a smooth function is Lipschitz continuous.

A global Holmgren theorem for multidimensional hyperbolic partial differential equations

Holmgrens theorem guarantees unique continuation across non-characteristic surfaces of class C 1 for solutions to homogeneous linear partial differential equations with analytic coefficients. Based on this result a global uniqueness theorem for solutions to hyperbolic differential equations with analytic coefficients in a space-time cylinder Q = (0, T) × Ω is established. Zero Cauchy data on a part of the lateral C 1-boundary will force every solution to a homogeneous hyperbolic PDE to vanish at half time T/2 provided that T > T 0. The minimal time T 0 depends on the geometry of the slowness surface of the hyperbolic operator, and can be determined explicitly as demonstrated by several examples including the system of transversely isotropic elasticity and Maxwells equations. Furthermore, it is shown that this global uniqueness result holds for hyperbolic operators with C 1-coefficients as long as they satisfy the conclusion of Holmgrens Theorem.

SemiGlobal Exact Controllability of Nonlinear Plates

The exact controllability problem for several semilinear thin plate models is considered. A distributed control affects a collar of the plates boundary. The result is semiglobal in the sense that there is no restriction on the size of the initial and the target states, but the controllability time is uniform only with respect to a given bounded set containing these states. Both (i) closed-loop-based and (ii) “pure” open-loop constructions are discussed. Strategy (i) describes exact controls for an abstract class of second-order evolution equations. It applies to the Berger plate with small in-plane stresses (and, depending on some open questions, possibly to the von Karman model). This method partly relies on uniform stabilization and offers no apparent leeway to improve the controllability time. Strategy (ii) is demonstrated on the example of a Kirchhoff model with a dissipative polynomial source term. Such a source serves as a prototype for ultimately considering the same control construction for the B...