Neumann Data Research Articles

Inverse Problems for One-Dimensional Fluid-Solid Interaction Models

AbstractWe consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one endpoint of the spatial interval. In particular, we establish unique results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities.

The 𝐿𝑝 restriction bounds for Neumann data on surface

Abstract Let { u λ } \{u_{\lambda}\} be a sequence of L 2 L^{2} -normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold ( M , g ) (M,g) . We seek to get an L p L^{p} restriction bound of the Neumann data λ − 1 ⁢ ∂ ν u λ | γ \lambda^{-1}\partial_{\nu}u_{\lambda}|_{\gamma} along a unit geodesic 𝛾. Using the 𝑇- T ∗ T^{*} argument, one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is O ⁢ ( λ − 1 p + 3 2 ) O(\lambda^{-\frac{1}{p}+\frac{3}{2}}) . The stationary phase theorem plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.

A Simplified Method for the Evaluation of Floating-Body Motion Responses over a Sloping Bottom

Recently, many floating renewable energy platforms have been deployed in coastal regions, where sloping bottoms are an important factor when evaluating their safety. In this article, a simplified method coupling an eigenfunction matching method (EMM) and a finite-depth Green’s function (FDGF) is developed to evaluate floating-body motion responses over a sloping bottom for which bathymetry is homogeneous in the longshore direction. We propose an extended EMM to create an incident wave model over the sloping bottom, thereby obtaining the Froude–Krylov (F–K) force and Neumann data on the wet surfaces of the floating body for the diffraction problem. An equivalent depth is introduced to account for the interaction between the sloping bottom and floating bodies when dealing with the diffraction and radiation problems. The accuracy of the present method is validated through a comprehensive comparison with numerical and/or experiment results for a liquefied natural gas (LNG) ship and a floating hemisphere from the literature. Generally, the present, simplified method can give predictions with sufficient accuracy.

An alternative potential method for mixed steady‐state elastic oscillation problems

We consider an alternative approach to investigate three‐dimensional exterior mixed boundary value problems (BVPs) for the steady‐state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, , has a compact boundary surface , which is divided into two disjoint parts, the Dirichlet part and the Neumann part , where the displacement vector (the Dirichlet‐type condition) and the stress vector (the Neumann‐type condition) are prescribed, respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed BVP in the form of a linear combination of the single‐layer and double‐layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov–Poincaré‐type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right‐hand sides of the resulting boundary integral equations system are vector functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate ‐based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector functions belonging to the Sobolev space and satisfying the Sommerfeld–Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the ‐based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses ‐Hölder continuity property in the closed domain with , where is an arbitrarily small number.

Free boundary problems of the incompressible Navier–Stokes equations with non-flat initial surface in the critical Besov space

Global well-posedness of the Navier–Stokes equations with a free boundary condition is considered in the scaling critical homogeneous Besov spaces B˙p,1-1+n/p(R+n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\dot{B}}_{p,1}^{-1+n/p}({\\mathbb {R}}^n_+)$$\\end{document} with n-1<p<2n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1< p< 2n-1$$\\end{document}. To show the global well-posedness, we establish end-point maximal L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-regularity for the initial-boundary value problem of the Stokes equations. Such an estimate is obtained via related estimate for the initial-boundary value problem of the heat equation with the inhomogeneous Neumann data as well as the pressure estimate in the critical Besov space framework. The proof heavily depends on the explicit expression of the fundamental integral kernel of the Lagrange transformed linearized Stokes equations and the almost orthogonal estimates with the space-time Littlewood–Paley dyadic decompositions. Our result here improves the initial space and boundary state than previous results by Danchin–Hieber–Mucha–Tolksdorf (Free boundary problems via Da Prato–Grisvard theory. arXiv:2011.07918v2) and ourselves (Ogawa and Shimizu in J Evol Equ 22(30):67, 2022; Ogawa and Shimizu in J Math Soc Jpn. arXiv:2211.06952v3).

A stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions for a 2D peridynamic diffusion model

In this work, a stable and high-order numerical scheme with discrete DtN-type artificial boundary conditions (ABCs) is designed for solving a peridynamic diffusion model on the whole two dimensional domain. To do so, we first use a high-order quadrature-based finite difference scheme to approximate the spatially nonlocal operator and use the BDF2 scheme to approximate the temporal direction. For the resulting fully discrete system, we apply the nonlocal potential theory to obtain the discrete Dirichlet-to-Dirichlet (DtD)-type ABCs. After that, we further derive the Dirichlet-to-Neumann (DtN)-type ABCs based on the nonlocal Neumann data defined from the discrete nonlocal Green's first identity. For the stability and convergence analysis, we reformulate the BDF2 operator into a discrete convolution sum form, then use the technique of the discrete orthogonal convolution kernels to obtain the optimal convergence order. Finally, numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

An inverse problem with partial Neumann data and $ L^{n/2} $ potentials

An inverse problem with partial Neumann data and $ L^{n/2} $ potentials

Homogenization and corrector result of optimal control problem for Stokes system

This article introduces the boundary optimal control problem in an N-dimensional domain governed by the stationary Stokes equations. Controls are applied to the states through Neumann data on the boundary. The aim of this paper is to study the asymptotic behavior of optimal control and states and identify the limit optimal control problem in the framework of the two-scale convergence. Finally, strong convergence of L 2 -cost functional and a corrector result for velocities is proved.

Determination of black holes by boundary measurements

For a wave equation with time-independent Lorentzian metric consider an initial-boundary value problem in [Formula: see text], where [Formula: see text] is the time variable and [Formula: see text] is a bounded domain in [Formula: see text]. Let [Formula: see text] be a subdomain of [Formula: see text]. We say that the boundary measurements are given on [Formula: see text] if we know the Dirichlet and Neumann data on [Formula: see text]. The inverse boundary value problem consists of recovery of the metric from the boundary measurements. In the author’s previous works a localized variant of the boundary control method was developed that allows the recovery of the metric locally in a neighborhood of any point of [Formula: see text] where the spatial part of the wave operator is elliptic. This allows the recovery of the metric in the exterior of the ergoregion. The goal of this survey paper is to recover the black holes. In some cases the ergoregion coincides with the black hole. In the case of two space dimensions we recover the black hole inside the ergoregion assuming that the ergosphere, i.e. the boundary of the ergoregion, is not characteristic at any point of the ergosphere.

Overdetermined boundary problems with nonconstant Dirichlet and Neumann data

Overdetermined boundary problems with nonconstant Dirichlet and Neumann data

Stability estimates for an inverse problem for Schrödinger operators at high frequencies from arbitrary partial boundary measurements

In this paper, we study the partial data inverse boundary value problem for the Schrödinger operator at a high frequency in a bounded domain with smooth boundary in , . Assuming that the potential is known in a neighborhood of the boundary, we obtain the logarithmic stability when both Dirichlet data and Neumann data are taken on arbitrary open subsets of the boundary where the two sets can be disjointed. Our results also show that the logarithmic stability can be improved to the one of Hölder type in the high frequency regime. To achieve those goals, we used a method by combining the CGO solution, Runge approximation and Carleman estimate.

Shape reconstruction for an inverse biharmonic problem from partial Cauchy data

In this paper, we address the Robin inverse problem for the biharmonic equation in a simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided.

A Tikhonov regularization method for Cauchy problem based on a new relaxation model

In this paper, we consider a Cauchy problem of recovering both missing value and flux on inaccessible boundary from Dirichlet and Neumann data measured on the remaining accessible boundary. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Compared to the existing work, weaker regularity is required on the Dirichlet data. This makes the proposed model simpler and more efficient in computation. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method. For simplicity of the statements, we take Poisson equation as the governed equation. However, the proposed method can be applied directly to Cauchy problems governed by more general equations, even other linear or nonlinear inverse problems.

A CCBM-based generalized GKB iterative regularization algorithm for inverse Cauchy problems

This paper examines inverse Cauchy problems that are governed by a kind of elliptic partial differential equation. The inverse problems involve recovering the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By using the coupled complex boundary method (CCBM), which integrates both Dirichlet and Neumann data into a single Robin boundary condition, we reformulate the underlying problem into an operator equation. Based on this new formulation, we study the solution existence issue of the reduced problem with noisy data. A Golub–Kahan bidiagonalization (GKB) process together with Givens rotation is employed for iteratively solving the proposed operator equation. The regularizing property of the developed method, called CCBM-GKB, and its convergence rate results are proved under a posteriori stopping rule. Finally, a linear finite element method is used for the numerical realization of CCBM-GKB. Various numerical experiments demonstrate that CCBM-GKB is a kind of accelerated iterative regularization method, as it is much faster than the classic Landweber method.

The Majda-Biello system on the half-line

The Majda-Biello system models the interaction of Rossby waves. It consists of two coupled KdV equations one of which has a parameter α as coefficient of its dispersion. This work studies this system on the half line with Robin, Neumann, and Dirichlet boundary data. It shows that for 0<α<1 or 1<α<4 all these problems are well-posed for initial data in Sobolev spaces Hs, s≥0. For α=1 or α>4 well-posedness holds for Dirichlet data if s>−3/4, while for Neumann and Robin data it depends on the sign of the parameters involved in the data. For α=4 well-posedness of all problems holds for s≥3/4. The Robin and Neumann boundary data are in Hs/3 while the Dirichlet boundary data are in H(s+1)/3. These are consistent with the time regularity of the Cauchy problem for the corresponding linear system. The proof is based on linear estimates in Bourgain spaces derived by utilizing the Fokas solution formula for the forced linear system, and appropriate bilinear estimates suggested by the coupled nonlinearities. These show that the iteration map defined via the Fokas formula is a contraction in appropriate solution spaces. All the well-posedness results obtained her e are optimal.

The enclosure method for the detection of variable order in fractional diffusion equations

&lt;p style='text-indent:20px;'&gt;This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of &lt;i&gt;the time domain enclosure method&lt;/i&gt; enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.&lt;/p&gt;

Wave-focusing to subsurface targets using a switching time-reversal mirror

The need for wave energy focusing to selected targets embedded within a host is commonly shared among various engineering fields, whether for stimulation and imaging in exploration geophysics, or for contaminant removal in geo-environmental engineering, or even for diagnostic/therapeutic purposes in medicine. In this paper, we are concerned with the feasibility of focusing wave energy to multiple subsurface targets embedded within a semi-infinite heterogeneous elastic host, by exploiting the time-reversibility of the lossless wave equation under adverse conditions for time reversal. Of particular interest for practical applications in geophysics is the need for a switching time-reversing mirror, where recorded Dirichlet data must be time-reversed as Neumann data due to equipment constraints. In the absence of theoretical guarantees, we turn to numerical experiments and report that wave-focusing can be indeed realized, despite various departures from the theoretically ideal time reversal case.

MIXED BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION

We consider a new approach to investigating the mixed boundary value problem (BVP) for the Helmholtz equation in the case of a three-dimensional unbounded domain Ω−⊂ℝ3 with compact boundary surface S=∂Ω−, which is divided into two disjoint parts, SD and SN, where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the classical potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP to a system of integral equations containing neither extensions of the Dirichlet or Neumann data nor a Steklov–Poincaré type operator involving the inverse of the single layer boundary integral operator, which is not available explicitly for arbitrary boundary surfaces. The right-hand sides of the resulting boundary integral equations system are functions coinciding with the given Dirichlet and Neumann data of the problem under consideration. We show that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the class of functions belonging to the Sobolev space W2, loc1(Ω−) and satisfying the Sommerfeld radiation conditions. We also show that the pseudodifferential matrix operator thus obtained is invertible in the Lp-based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses the Cα-Hölder continuity property in the closed domain Ω¯− with α=12−𝜀, where 𝜀>0 is an arbitrarily small number.

The Dirichlet-conormal problem for the heat equation with inhomogeneous boundary conditions

We consider the mixed Dirichlet-conormal problem for the heat equation on cylindrical domains with a bounded and Lipschitz base Ω⊂Rd and a time-dependent separation Λ. Under certain mild regularity assumptions on Λ, we show that for any q>1 sufficiently close to 1, the mixed problem in Lq is solvable. In other words, for any given Dirichlet data in the parabolic Riesz potential space Lq1 and the Neumann data in Lq, there is a unique solution and the non-tangential maximal function of its gradient is in Lq on the lateral boundary of the domain. When q=1, a similar result is shown when the data is in the Hardy space. Under the additional condition that the boundary of the domain Ω is Reifenberg-flat and the separation is locally sufficiently close to a Lipschitz function of m variables, where m=0,…,d−2, with respect to the Hausdorff distance, we also prove the unique solvability result for any q∈(1,(m+2)/(m+1)). In particular, when m=0, i.e., Λ is Reifenberg-flat of co-dimension 2, we derive the Lq solvability in the optimal range q∈(1,2). For the Laplace equation, such results were established in [25,24,5] and [13].

Propagation of smallness and size estimate in the second order elliptic equation with discontinuous complex Lipschitz conductivity

In this paper, we would like to derive three-ball inequalities and propagation of smallness for the complex second order elliptic equation with discontinuous Lipschitz coefficients. As an application of such estimates, we study the size estimate problem by one pair of Cauchy data on the boundary, that is, a pair of the Neumann and Dirichlet data of the solution on the boundary. The main ingredient in the derivation of three-ball inequalities and propagation of smallness is a local Carleman proved in our recent paper [13].