Conditions For Well-posedness Research Articles

Nature of instability in flow-driven porous anodic oxide.

Self-organized porous anodic oxide films are formed by the electrochemical oxidation of reactive metal aluminum in acidic solutions in which the oxide is soluble. Recently, viscous flow models have shown using linear stability analysis that the instability results from a trade-off between the destabilizing effect of viscous flow of oxide and the stabilizing effect of oxide formation, which provides the wavelength selection mechanism for pattern formation. Anion adsorption on surface growth sites causes nonuniform compressive stress at the oxide-solution interface, which drives the flow. This anodic instability is analogous to the classical Marangoni instability induced by surface tension gradients. In this work, nature of the instability beyond the stability threshold is determined using a weakly nonlinear analysis. For the growth of well-developed pores beyond the threshold, a subcritical nature of the instability is essential. However, our weakly nonlinear analysis shows that the solutions emerging from neutral stability are supercritical in nature at all wavenumbers for the practical range of anodizing control parameters investigated. We also determine the region where the model is Hadamard stable, a necessary condition for well-posedness.

On the Cauchy problem for p-evolution equations with variable coefficients: A necessary condition for Gevrey well-posedness

In this paper we consider a class of p-evolution equations of arbitrary order with variable coefficients depending on time and space variables (t,x). We prove necessary conditions on the decay rates of the coefficients for the well-posedness of the related Cauchy problem in Gevrey spaces.

On well-posedness of Navier–Stokes variational inequalities

In this paper, well-posedness is carried out on variational inequalities in a general form for the stationary Navier–Stokes equations modeling viscous incompressible fluid flows with a nonsmooth slip boundary condition of friction type. For a special form of the variational inequality commonly studied in existing literature, our proof technique leads to a substantial improvement on the well-posedness condition.

Well-posedness characterizations for system of mixed hemivariational inequalities

In this paper, we concern with the concept of well-posedness and well-posedness in generalized sense for a system of mixed hemivariational inequality (SMHVI) with perturbations. We establish several metric characterizations and equivalent conditions of well-posedness for SMHVI. Our main results improve and extend some announced work in the literature.

Integral Equations of Volterra Typewith Two Boundary and One Interior Singular Point

Explicit solutions of model and nonmodel integral equations of Volterra type with two boundary and one interior singular point are obtained, and the properties of the resulting solutions are studied. The well-posed statement of problems with conditions specified on singular manifolds is found in the case where the solution of the model equation contains an arbitrary constant.

Well-posedness and properties of the flow for semilinear evolution equations

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers’ equation with nonlinear local dynamics and both distributed and boundary disturbances.

Boundary feedback stabilization of an unstable cascaded heat–heat system with different reaction coefficients

The present paper addresses a boundary stabilization problem for a cascaded heat–heat system with different reaction coefficients. A sufficient and necessary condition about two reaction coefficients of the cascaded heat–heat model is found to guarantee the controllability of the coupled system. Based on modal decomposition method, a feedback control is proposed to stabilize the coupled system. Sufficient conditions for exponential stability and well-posedness of the closed-loop systems are derived via the Lyapunov method. The performance is validated by the numerical simulation, which illustrates the efficiency of the proposed control scheme.

Analysis of nonlinear poroviscoelastic flows with discontinuous porosities *

Abstract Existence and uniqueness of solutions is shown for a class of viscoelastic flows in porous media with particular attention to problems with nonsmooth porosities. The considered models are formulated in terms of the time-dependent nonlinear interaction between porosity and effective pressure, which in certain cases leads to porosity waves. In particular, conditions for well-posedness in the presence of initial porosities with jump discontinuities are identified.

Local well-posedness and standing waves with prescribed mass for Schrodinger-Poisson systems with a logarithmic potential in R^2

In this article, we consider planar Schrodinger-Poisson systems with a logarithmic external potential \(W(x)=\ln (1+|x|^2)\) and a general nonlinear term \(f\). We obtain conditions for the local well-posedness of the Cauchy problem in the energy space. By introducing some suitable assumptions on \(f\), we prove the existence of the global minimizer. In addition, with the help of the local well-posedness, we show that the set of ground state standing waves is orbitally stable.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/64/abstr.html

Conditions of well-posedness for planar conewise linear systems

A planar (2D) conewise linear system (CLS) is considered. This is a piecewise linear system of two states and multiple modes, where each mode is linear with its state-space constrained into a polyhedral, finitely generated, convex cone. It is allowed to have a discontinuous vector field and sliding modes. Alternative conditions for well-posedness of Caratheodory solutions of this system that have intuitive interpretations with respect to eigenvectors and cone-boundary vectors are derived. It is also shown that a well-known condition for well-posedness of bimodal systems also applies to two adjacent modes of this system without any change.

Solving Nonlinear Algebraic Loops Arising in Input-Saturated Feedbacks

We propose a dynamic augmentation scheme for the asymptotic solution of the nonlinear algebraic loops arising in well-known input saturated feedbacks typically designed by solving linear matrix inequalities (LMIs). We prove that the existing approach based on dynamic augmentation, which replaces the static loop by a dynamic one through the introduction of a sufficiently small time constant, works under some restrictive sufficient well-posedness conditions, requiring the existence of a diagonal Lyapunov matrix. However it can fail in general, even when the algebraic loop is well-posed. Then, we propose a novel approach whose effectiveness is guaranteed whenever well-posedness holds. We also show how this augmentation allows preserving the guaranteed region of attraction with Lyapunov-based designs, as long as a gain parameter is sufficiently large. We finally propose an adaptive version of the scheme where this parameter is adjusted online. Simulation results show the effectiveness of the proposed solutions.

Portfolio selection with exploration of new investment assets

We introduce a model for portfolio selection with an extendable investment universe where an agent with mean-variance preferences faces a trade-off between exploiting existing and exploring for new investment opportunities. When the agent chooses to explore, a new risky asset is discovered and the agent subsequently invests in the extended universe. We first provide conditions for wellposedness and characterize the optimal amount devoted to exploration. Our model shows that incremental exploration does not pay off, that it is increasingly worthwhile to explore in worse market environments, and that investment performance measured by the Sharpe ratio is increasing in the initial wealth of the agent indicating that richer agents can make better use of new investment opportunities. We further generalize our model and verify the robustness of the main findings with regard to various modeling assumptions.

Sensitivity Analysis of Stochastic Constraint and Variational Systems via Generalized Differentiation

This paper conducts sensitivity analysis of random constraint and variational systems related to stochastic optimization and variational inequalities. We establish efficient conditions for well-posedness, in the sense of robust Lipschitzian stability and/or metric regularity, of such systems by employing and developing coderivative characterizations of well-posedness properties for random multifunctions and efficiently evaluating coderivatives of special classes of random integral set-valued mappings that naturally emerge in stochastic programming and stochastic variational inequalities.

Обобщение метода Петрова–Галеркина для решения системы интегральных уравнений Фредгольма второго рода

The numerical solution problem of a second kind Fredholm m integral equations linear system is considered. The Petrov–Galerkin projection method generalization for solving this problem is proposed the first time. n is a coordinate functions number of two linearly independent systems. It can be equal, more or less then the m – the integral equations number. This algorithm advantage is that it is not sensitive to the parameters λ smallness in the integral equations system. The algorithm requires the correct choice of two linearly independent coordinate functions systems and their number. The solution algorithm is reduced to a matrix solution for an antidiagonal problem. Two examples are solved for the antidiagonal problem, and numerical solutions of the problem coincide with the exact solutions. Two theorems are proved for sufficient conditions for the proposed numerical algorithms correctness in two cases. In the first case, an antidiagonal problem is considered with two Fredholm equations of the second kind. The second theorem considers well-posedness conditions for a general diagonal problem. Undoubtedly, the proposed algorithm will be useful in mechanics and computational mathematics problems.

On bounded solutions of linear systems of differential equations with unbounded coefficients

This paper deals with a problem of finding a bounded solution of a system of nonhomogeneous linear differential equations with an unbounded matrix of coefficients on a finite interval. The right-hand side of the equation belongs to a space of continuous functions bounded with some weight; the weight function is chosen taking into account the behavior of the coefficient matrix. The problem is studied using a modified version of the parameterization method with non-uniform partitioning. Necessary and sufficient conditions of well-posedness of the problem are obtained in terms of a bilaterally infinite matrix of special structure.

Partially dissipative hyperbolic systems in the critical regularity setting: The multi-dimensional case

We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole space Rd with d≥2. Following our recent work dedicated to the one-dimensional case [11], we establish the existence of global strong solutions and decay estimates in the critical regularity setting whenever the system under consideration satisfies the so-called (SK) (for Shizuta-Kawashima) condition. Our results in particular apply to the compressible Euler system with damping in the velocity equation.Compared to the recent works devoted to similar issues, our use of hybrid Besov norms with different regularity exponents in low and high frequency enables us to pinpoint optimal smallness conditions for global well-posedness and to get a more accurate information on the qualitative properties of the constructed solutions.A great part of our analysis relies on the study of a Lyapunov functional in the spirit of that of Beauchard and Zuazua in [2]. Exhibiting a damped mode with faster time decay than the whole solution also plays a key role.

Well-posedness for set-valued equilibrium problems

"In this paper we extend a concept of well-posedness for vector equilibrium problems to the more general framework of set-valued equilibrium problems in topological vector spaces using an appropriate reformulation of the concept of minimality for sets. Su cient conditions for well-posedness are given in the generalized convex settings and we are able to single out classes of well-posed set-valued equilibrium problems. On the other hand, in order to relax some conditions, we introduce a concept of minimizing sequences for a set-valued problem, in the set criterion sense, and further we will have a concept of well-posedness for the set-valued equilibrium problem we are interested in. Suficient results are also given for this well-posedness concept."

First-Kind Boundary Integral Equations for the Dirac Operator in 3-Dimensional Lipschitz Domains

We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Garding inequalities are derived and we establish that the obtained boundary integral operators are Fredholm of index zero. Their finite dimensional kernels are characterized and we show that their dimension is equal to the number of topological invariants of the domain's boundary, in other words to the sum of its Betti numbers. This is explained by the fundamental discovery that the associated bilinear forms agree with those induced by the 2D surface Dirac operators for H-1/2 surface de Rham Hilbert complexes whose underlying inner-products are the non-local inner products defined through the classical single-layer boundary integral operators for the Laplacian. Decay conditions for well-posedness in natural energy spaces of the Dirac system in unbounded exterior domains are also presented.

Well-Posedness, Internal Stability and Input-Output Stability in Networked Multi-Agent Systems

In this article, we consider a feedback interconnection, which consists of a networked multi-agent system with its distributed controller and two sets of external input signals. Assuming that the agents are identical, and the dynamics of each agent and its local controller are represented by transfer functions, the well-posedness condition, internal stability and input-output stability of the feedback interconnection are investigated. A controller is designed by solving a Diophantine equation, and a gain for the network is selected using root-locus of a unity feedback system to achieve input-output stability. The developed results are demonstrated with a numerical example.

A data-driven peridynamic continuum model for upscaling molecular dynamics

Nonlocal models, including peridynamics, often use integral operators that embed lengthscales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials’ behavior, we propose a learning framework to extract, from MD data, an optimal Linear Peridynamic Solid (LPS) model as a surrogate for MD displacements. To maximize the accuracy of the learnt model we allow the peridynamic influence function to be partially negative, while preserving the well-posedness of the resulting model. To achieve this, we provide sufficient well-posedness conditions for discretized LPS models with sign-changing influence functions and develop a constrained optimization algorithm that minimizes the equation residual while enforcing such solvability conditions. This framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene. Our two-dimensional tests show the robustness of the proposed algorithm on validation data sets that include thermal noise, different domain shapes and external loadings, and discretizations substantially different from the ones used for training.