Perturbations Of Initial Data Research Articles

Insensitizing controls for the micropolar fluids

In this paper, we investigate the existence of insensitizing controls for the micropolar fluids in a bounded domain with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. The study of insensitizing controls is essential to solve a stability problem, which means that we look for controls such that some functionals of the velocity fields (the so‐called sentinels) are insensitive to the small perturbations of initial data. The problem of insensitizing controls is transformed into a suitable controllability problem for a cascade system. Our proof relies on a new global Carleman inequality and the inverse mapping theorem.

Insensitizing control problem for the Hirota–Satsuma system of KdV–KdV type

This paper is concerned with the existence of insensitizing controls for a nonlinear coupled system of two Korteweg–de Vries (KdV) equations, typically known as the Hirota–Satsuma system. The idea is to look for controls such that some functional of the states (the so-called sentinel) is insensitive to the small perturbations of initial data. Since the system is coupled, we consider a sentinel in which we observe both components of the system in a localized observation set. By some classical argument, the insensitizing problem is then reduced to a null-control problem for an extended system where the number of equations is doubled. We study the null-controllability for the linearized model associated to that extended system by means of a suitable Carleman estimate which is proved in this paper. Finally, the local null-controllability of the extended (nonlinear) system is obtained by applying the inverse mapping theorem, and this implies the required insensitizing property for the concerned model.

Convergence to traveling waves in reaction-diffusion systems with equal diffusivities

In this paper, we first derive a theorem on the convergence of solutions to traveling waves in reaction-diffusion systems with equal diffusivities. Then we apply this theorem to some specific examples of predator-prey models which were studied recently in the literature. This gives a stability result for these traveling waves in the corresponding predator-prey system under certain perturbations of initial data.

Asymptotics of a Solution to an Optimal Control Problem with Integral Convex Performance Index, Cheap Control, and Initial Data Perturbations

Asymptotics of a Solution to an Optimal Control Problem with Integral Convex Performance Index, Cheap Control, and Initial Data Perturbations

On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime

The non-linear stability of the sub-extremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary is analysed using a technique based on the extended conformal Einstein field equations and a conformal Gaussian gauge. This strategy relies on the observation that the Cosmological stationary region of this exact solution can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of suitable spacelike hypersurfaces in the Cosmological region of the spacetime can be expressed in terms of a conformal Gaussian gauge. A perturbative argument then allows to prove existence and stability results close to the conformal boundary and away from the asymptotic points where the Cosmological horizon intersects the conformal boundary. In particular, we show that small enough perturbations of initial data for the sub-extremal Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which is regular at the conformal boundary. The analysis in this article can be regarded as a first step towards a stability argument for perturbation data on the Cosmological horizons.

Stability of self-similar solutions to geometric flows

We show that self-similar solutions for the mean curvature flow, surface diffusion, and Willmore flow of entire graphs are stable upon perturbations of initial data with small Lipschitz norm. Roughly speaking, the perturbed solutions are asymptotically self-similar as time tends to infinity. Our results are built upon the global analytic solutions constructed by Koch and Lamm in 2012, the compactness arguments adapted by Asai and Giga in 2014, and the spatial equi-decay properties on certain weighted function spaces. The proof for all of the above flows are achieved in a unified framework by utilizing the estimates of the linearized operator.

Riemann problem and wave interactions for an inhomogeneous Aw-Rascle traffic flow model with extended Chaplygin gas

The Riemann problem and wave interactions are discussed and investigated for an inhomogeneous Aw-Rascle (AR) traffic flow model with extended Chaplygin gas pressure. First, under some variable transformation, the Riemann problem with initial data of two piecewise constants is solved and two different types of Riemann solutions involving rarefaction wave, shock wave and contact discontinuity are obtained. Second, by studying the Riemann problem with three-piecewise-constant initial data, we analyze the interactions of waves and establish the global structures of Riemann solutions. It is shown that, influenced by the source term, the Riemann solutions for the inhomogeneous AR traffic flow model are no longer self-similar, and all the elementary wave curves do not keep straight. Finally, the stability of solution under the small perturbation of initial data is briefly discussed.

About stability of mixed integer optimization problems under perturbations of input data of vector criterion

The article is devoted to the study of the influence of uncertainty in initial data on the solutions of mixed integer optimization vector problems. In the optimization problems, including problems with vector criterion, small perturbations in initial data can result in solutions strongly different from the true ones. The problem of stability of the indicated tasks is studied from the point of view of direct coupled with her question in relation to stability of solutions belonging to some subsets of feasible set.

Stability kernel of vector optimization problems under perturbations of criterion functions

The article is devoted to the study of the influence of uncertainty in initial data on the solutions of optimization multicriterial problems. In the optimization problems, including problems with vector criterion, small perturbations in initial data can result in solutions strongly different from the true ones. The results of the conducted researches allow us to extend the known class of vector optimization problems, stable with respect to input data perturbations in vector criterion. We are talking about stability in the sense of Hausdorff lower semicontinuity for point-set mapping that characterizes the dependence of the set of optimal solutions on the input data of the vector optimization problem. The conditions of stability against input data perturbations in vector criterion for the problem of finding Pareto optimal solutions with continuous partial criterion functions and feasible set of arbitrary structure are obtained by studying the sets of points that are stable belonging and stable not belonging to the Pareto set.

An update on combinatorial method for aggregation of expert judgments in AHP

Paper aims the paper aims to demonstrate the advantages of several modifications of combinatorial method of expert judgment aggregation in AHP. Modifications are based on 1) weighting of spanning trees; 2) sorting of spanning trees by graph diameter during aggregation. Originality Both the method and its modifications are developed and improved by the authors. We propose to 1) weight spanning trees, based on quality of respective expert estimates, and 2) sort them by diameter in order to reduce the impact of expert errors and the method’s computational complexity. Research method we focus on theoretical and empirical studies of several modifications of combinatorial method of aggregation of expert judgments within AHP. Main findings modified combinatorial method has several conceptual advantages over ordinary method. It is also less sensitive to perturbations of initial data. Additionally, selection of spanning trees with smaller diameters allows us to reduce computational complexity of the method and minimize the impact of expert errors. Implications for theory and practice Combinatorial method is a universal instrument of expert judgment aggregation, applicable to additive/multiplicative, complete/incomplete, individual/group pair-wise comparisons, provided in different scales. It is used in the original strategic planning technology, which has recently found several important applications.

Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations

<p style='text-indent:20px;'>We study the large-time asymptotic behavior of solutions toward the rarefaction wave of the compressible non-isentropic Navier-Stokes equations coupling with Maxwell equations under some small perturbations of initial data and also under the assumption that the dielectric constant is bounded. For that, the dissipative structure of this hyperbolic-parabolic system is studied to include the effect of the electromagnetic field into the viscous fluid and turns out to be more complicated than that in the simpler compressible Navier-Stokes system. The proof of the main result is based on the elementary <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> energy methods.

Stability of rarefaction wave for isentropic compressible Navier–Stokes–Maxwell equations

We study the large-time asymptotic behavior of solutions toward the weak rarefaction wave of Navier–Stokes equations coupling with the Maxwell equations under small perturbations of initial data and also under the dielectric constant suitably small. The result is proved based on elementary L2 energy methods.

On Global Dynamics in Duffing Equation with Quasiperiodic Perturbation

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

On an iterative method for solution of direct problem for nonlinear hyperbolic differential equations

An iterative method for solution of Cauchy problem for one-dimensional nonlinear hyperbolic differential equation is proposed in this paper. The method is based on continuous method for solution of nonlinear operator equations. The keystone idea of the method consists in transition from the original problem to a nonlinear integral equation and its successive solution via construction of an auxiliary system of nonlinear differential equations that can be solved with the help of different numerical methods. The result is presented as a mesh function that consists of approximate values of the solution of stated problem and is constructed on a uniform mesh in a bounded domain of two-dimensional space. The advantages of the method are its simplicity and also its universality in the sense that the method can be applied for solving problems with a wide range of nonlinearities. Finally it should be mentioned that one of the important advantages of the proposed method is its stability to perturbations of initial data that is substantiated by methods for analysis of stability of solutions of systems of ordinary differential equations. Solving several model problems shows effectiveness of the proposed method.

Stability Studies of a Coupled Model to Perturbation of Initial Data

The stability problem is considered in terms of the classical Lyapunov definition. For this, a set of initial conditions is set, consisting of their preliminary calculations, and the spread of the trajectories obtained as a result of numerical simulation is analyzed. This procedure is implemented as a series of ensemble experiments with a joint MPI-ESM model of the Institute of Meteorology M. Planck (Germany). For numerical modeling, a series of different initial values of the characteristic fields was specified and the model was integrated, starting from each of these fields for different time periods. Изучались экстремальные характеристики уровня океана за период 30 лет. The statistical distribution was built, the parameters of this distribution were estimated, and the statistical forecast for 5 years in advance was studied. It is shown that the statistical forecast of the level corresponds to the calculated forecast obtained by the model. The localization of extreme level values was studied and an analysis of these results was carried out. Numerical calculations were performed on the Lomonosov-2 supercomputer of Lomonosov Moscow State University.

Spectral methods to study the robustness of residual neural networks with infinite layers

Recently, neural networks (NN) with an infinite number of layers have been introduced. Especially for these very large NN the training procedure is very expensive. Hence, there is interest to study their robustness with respect to input data to avoid unnecessarily retraining the network. Typically, model-based statistical inference methods, e.g. Bayesian neural networks, are used to quantify uncertainties. Here, we consider a special class of residual neural networks and we study the case, when the number of layers can be arbitrarily large. Then, kinetic theory allows to interpret the network as a dynamical system, described by a partial differential equation. We study the robustness of the mean-field neural network with respect to perturbations in initial data by applying UQ approaches on the loss functions.

Approximation and Existence of Vacuum States in the Multiscale Flows of Compressible Euler Equations

In this paper, we study the approximation and existence of vacuum states in the multiscale gas flows governed by the Cauchy problem of compressible Euler equations containing a small parameter $\eta$ in the initial density. The system of compressible Euler equations is reduced to a hyperbolic resonant system at the vacuum so that the weak solution of the Riemann problem is not suitable as the building block of the Glimm (or Godunov) scheme to establish the existence of vacuum states. We construct a new type of approximate solutions, the weak solutions of the regularized Riemann problem for the leading-order system derived from asymptotic expansions around vacuum states. Such an approximate solution obtained by solving the pressureless Euler equations with generalized Riemann data consists of constant states separated by a composite hyperbolic wave. We show the stability of the regularized Riemann solution, together with the numerical simulations, under the small perturbations of initial data. Adopting the approximate solution as the building block of the generalized Glimm scheme, we prove the existence of the vacuum states by showing the stability and consistency of the scheme as $\eta \rightarrow 0$. The numerical simulation indicates that for any small fixed $t>0$, the approximate solutions converge to the exact solutions of the Cauchy problem in $L^1$ as $\eta \rightarrow 0$. The results of this paper can be applied to some hyperbolic resonant systems of balance laws.

The Riemann problem with delta initial data for the two-dimensional steady zero-pressure adiabatic flow

For the two-dimensional steady zero-pressure adiabatic flow, the Riemann problem with delta initial data is investigated and the global existence of generalized solution is established in four cases. Particularly, in solutions, a special type of nonclassical wave called the delta contact discontinuity with Dirac delta functions developing in both state variables is found. Furthermore, we show that the constructed generalized solutions are stable by the perturbation of initial data.

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

AbstractIn this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu&gt;0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q&gt;1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q&gt;0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].

Construction of a stable blowup solution with a prescribed behavior for a non-scaling-invariant semilinear heat equation

We consider the semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + |u|^{p-1} u \ln ^{\alpha}( u^2 +2), \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $p > 1$ and $ \alpha \in \mathbb{R}$. Unlike the standard case $\alpha = 0$, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time $T$ only at one blowup point $a$, according to the following asymptotic dynamics: \begin{eqnarray*} u(x,t) \sim \psi(t) \left(1 + \frac{(p-1)|x-a|^2}{4p(T -t)|\ln(T -t)|} \right)^{-\frac{1}{p-1}} \text{ as } t \to T, \end{eqnarray*} where $\psi(t)$ is the unique positive solution of the ODE \begin{eqnarray*} \psi' = \psi^p \ln^{\alpha}(\psi^2 +2), \quad \lim_{t\to T}\psi(t) = + \infty. \end{eqnarray*} The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finite dimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely non-scale invariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.