Solution Map Research Articles

Nowhere-differentiability of the solution map of 2D Euler equations on bounded spatial domain

We consider the incompressible 2D Euler equations on bounded spatial domain $S$, and study the solution map on the Sobolev spaces $H^k(S)$ ($k > 2$). Through an elaborate geometric construction, we show that for any $T >0$, the time $T$ solution map $u_0 \mapsto u(T)$ is nowhere locally uniformly continuous and nowhere Frechet differentiable.

On the regularity of the solution map of the Euler-Poisson system

In this paper we consider the Euler--Poisson system (describing a plasma consisting of positive ions with a negligible temperature and massless electrons in thermodynamical equilibrium) on the Sobolev spaces $H^s(\mathbb{R}^3)$, $s > 5/2$. Using a geometric approach we show that for any time $T > 0$ the corresponding solution map, $(\rho_0,u_0) \mapsto (\rho(T),u(T))$, is nowhere locally uniformly continuous. On the other hand it turns out that the trajectories of the ions are analytic curves in $\mathbb{R}^3$.

Sensitivity Analysis and Optimal Control of Obstacle-Type Evolution Variational Inequalities

This paper is concerned with the differential sensitivity analysis and the optimal control of evolution variational inequalities (EVIs) of obstacle type. We demonstrate by means of a counterexample that the solution map $S$ of an EVI with a unilateral constraint is typically not (weakly) directionally differentiable or Lipschitz continuous in any of the spaces $H^s(0, T; H)$, $s \geq 1/2$, where $(0, T)$ is the time interval and $H$ is the pivot space of the underlying Gelfand triple $V \hookrightarrow H \hookrightarrow V^*$. We further establish that, despite this negative result, the solution operator is always strongly Hadamard directionally differentiable as a function $S : L^2(0, T; H) \to L^q(0, T; H)$ for all $1 \leq q < \infty$, weakly-$\star$ directionally differentiable as a function $S : L^2(0, T; H) \to L^\infty(0, T; H)$, and weakly directionally differentiable as a function $S : L^2(0, T; H) \to L^2(0, T; V)$. Using the differentiability properties of the map $S$, we derive strong stationarity conditions for optimal control problems that are governed by EVIs of obstacle type. The resulting optimality system is compared with that obtained by regularization.

Canonical RDEs and general semimartingales as rough paths

In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of L\'epingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.

A Method of Directly Defining the inverse Mapping (MDDiM) for solutions of non-linear coupled systems arising in SIR and SIS epidemic models

A Method of Directly Defining the inverse Mapping (MDDiM) for solutions of non-linear coupled systems arising in SIR and SIS epidemic models

Mathematical modeling and analysis of harmful algal blooms in flowing habitats.

In this paper, we survey recent developments of mathematical modeling and analysis of the dynamics of harmful algae in riverine reservoirs. To make the models more realistic, a hydraulic storage zone is incorporated into a flow reactor model and new mathematical challenges arise from the loss of compactness of the solution maps. The key point in the study of the evolution dynamics is to prove the existence of global attractors for the model systems and the principal eigenvalues for the associated linearized systems without compactness.

Moments and regularity for a Boltzmann equation via Wigner transform

In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $ \mathbb{R}^d $, $ d\geq 2 $, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when $ \left&lt; v \right&gt;^\beta f_0 \in L^2_v H^\alpha_x $ for $ \min (\alpha,\beta) &gt; \frac{d-1}{2} $. We prove that if $ \alpha,\beta $ are large enough, then it is possible to propagate moments in $ x $ and derivatives in $ v $ (for instance, $ \left&lt; x \right&gt;^k \left&lt; \nabla_v \right&gt;^\ell f \in L^\infty_T L^2_{x,v} $ if $ f_0 $ is nice enough). The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of $ f $. We also prove a persistence of regularity result for the scale of Sobolev spaces $ H^{\alpha,\beta} $; and, continuity of the solution map in $ H^{\alpha,\beta} $. Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the $ H $-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in $ H^{\alpha,\beta} $ for any $ \alpha,\beta &gt; \frac{d-1}{2} $, without any additional regularity or decay assumptions.

Approximation of the interface condition for stochastic Stefan-type problems

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.

Lower semicontinuity of approximate solution mappings for a parametric generalized strong vector equilibrium problem

In this paper, by using a property of convex mappings, one establishes the lower semicontinuity of the approximate solution mapping to a parametric generalized strong vector equilibrium problem without these assumptions about monotonicity and compactness. Our proof approach is different from the ones in the literature.

Time-varying delays in electrophysiological wave propagation along cardiac tissue and minimax control problems associated with uncertain bidomain type models

Motivated by topics and issues critical to human health, the problem studied in this work derives from the modeling and stabilizing control of electrical cardiac activity in order to maximize the efficiency and safety of treatment for cardiac disease. In this paper we consider nonlinear minimax control problems constrained by an uncertain modified bidomain model of cardiac tissue electrophysiology system, in order to take into account the influence of noises in data and time-delays in signal transmission. The state system is a degenerate nonlinear coupled system of reaction-diffusion equations in the shape of a set of delay differential equations coupled with a set of delay partial differential equations with multiple time-varying delays. The concept of our minimax control approach consists in setting the problem in the worst-case disturbances which leads to the game theory in which the controls and disturbances play antagonistic roles. The proposed strategy consists in controlling these instabilities by acting on certain data to maintain the system in a desired state. First, the mathematical model is introduced and its well-posedness is studied. Second, the minimax control problem is formulated. Afterwards the Frechet differentiability of nonlinear solution map from the couple control-disturbance input to the solution of state system is assessed as well as stability of the derived sensitive system. The existence of an optimal solution is proved and first-order necessary optimality conditions are established by using sensitivity and adjoint calculus.

Second-Order Optimality Conditions and Solution Stability to Optimal Control Problems Governed by Stationary Navier-Stokes Equations

In this paper, a class of parametric optimal control problems governed by stationary Navier-Stokes equations with mixed pointwise constraints is considered. We give no-gap second-order necessary and sufficient conditions for unperturbed problem. We show that if the strictly second-order sufficient condition for unperturbed problem is valid and the objective function is locally Lipschitz continuous, then the solution map is locally upper Holder continuous at the reference parameter.

L0 -Motivated Low-Rank Sparse Subspace Clustering.

In many applications, high-dimensional data points can be well represented by low-dimensional subspaces. To identify the subspaces, it is important to capture a global and local structure of the data which is achieved by imposing low-rank and sparseness constraints on the data representation matrix. In low-rank sparse subspace clustering (LRSSC), nuclear and l1 -norms are used to measure rank and sparsity. However, the use of nuclear and l1 -norms leads to an overpenalized problem and only approximates the original problem. In this paper, we propose two l0 quasi-norm-based regularizations. First, this paper presents regularization based on multivariate generalization of minimax-concave penalty (GMC-LRSSC), which contains the global minimizers of a l0 quasi-norm regularized objective. Afterward, we introduce the Schatten-0 ( S0 ) and l0 -regularized objective and approximate the proximal map of the joint solution using a proximal average method ( S0/l0 -LRSSC). The resulting nonconvex optimization problems are solved using an alternating direction method of multipliers with established convergence conditions of both algorithms. Results obtained on synthetic and four real-world datasets show the effectiveness of GMC-LRSSC and S0/l0 -LRSSC when compared to state-of-the-art methods.

Stability analysis for generalized semi-infinite optimization problems under functional perturbations

The concepts of essential solutions and essential solution sets for generalized semi-infinite optimization problems (GSIO for brevity) are introduced under functional perturbations, and the relations among the concepts of essential solutions, essential solution sets and lower semicontinuity of solution mappings are discussed. We show that a solution is essential if and only if the solution is unique; and a solution subset is essential if and only if it is the solution set itself. Some sufficient conditions for the upper semicontinuity of solution mappings are obtained. Finally, we show that every GSIO problem can be arbitrarily approximated by stable GSIO problems (the solution mapping is continuous), i.e., the set of all stable GSIO problems is dense in the set of all GSIO problems with the given topology.

5G D2D Networks: Techniques, Challenges, and Future Prospects

The increasing number of mobile users has given impetus to the demand for high data rate proximity services. The fifth-generation (5G) wireless systems promise to improve the existing technology according to the future demands and provide a road map for reliable and resource-efficient solutions. Device-to-device (D2D) communication has been envisioned as an allied technology of 5G wireless systems for providing services that include live data and video sharing. A D2D communication technique opens new horizons of device-centric communications, i.e., exploiting direct D2D links instead of relying solely on cellular links. Offloading traffic from traditional network-centric entities to D2D network enables low computational complexity at the base station besides increasing the network capacity. However, there are several challenges associated with D2D communication. In this paper, we present a survey of the existing methodologies related to aspects such as interference management, network discovery, proximity services, and network security in D2D networks. We conclude by introducing new dimensions with regard to D2D communication and delineate aspects that require further research.

Continuity of Solutions for Parametric Set Optimization Problems via Scalarization Methods

The aim of this paper is to investigate the continuity of solution mappings for parametric set optimization problems with upper and lower set less order relations by scalarization methods. First, we recall some linear and nonlinear scalarization properties used to characterize the set order relations. Subsequently, we introduce new monotonicity concepts of the set-valued mapping by linear and nonlinear scalarization methods. Finally, we obtain the semicontinuity and closedness of solution mappings for parametric set optimization problems (both convex and nonconvex cases) under the monotonicity assumption and other suitable conditions. The results achieved do not impose the continuity of the set-valued objective mapping, which are obviously different from the related ones in the literature.

On the R0-Tensors and the Solution Map of Tensor Complementarity Problems

Our purpose is to investigate the local boundedness, the upper semicontinuity, and the stability of the solution map of tensor complementarity problems. To do this, we focus on the set of R$_0$--tensors and show that this set plays an important role in the investigation. Furthermore, by using a technique in semi-algebraic geometry, we obtain some results on the finite-valuedness and the lower semicontinuity of the solution map.

A note on semicontinuity of the solution mapping for parametric set optimization problems

The present work is devoted to studying the stability of a parametric set optimization problem. In particular, based on the partial order relation on the family of nonempty bounded sets defined by Karaman et al. (Positivity 22(3):783–802, 2018), we give some sufficient conditions for the upper semicontinuity, lower semicontinuity, and closedness of the solution mapping to a parametric set optimization problem.

The Cauchy problem for a generalized Camassa–Holm equation

In this paper, we are concerned with the Cauchy problem of the generalized Camassa–Holm equation. Using a Galerkin-type approximation scheme, it is shown that this equation is well-posed in Sobolev spaces Hs, s>3/2 for both the periodic and the nonperiodic case in the sense of Hadamard. That is, the data-to-solution map is continuous. Furthermore, it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous. The nonuniform dependence is proved using the method of approximate solutions and well-posedness estimates. Moreover, it is shown that the solution map for the generalized Camassa–Holm equation is Hölder continuous in Hr-topology. Finally, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time.

Levitin–Polyak well-posedness for set optimization problems involving set order relations

In this paper set optimization problems with three types of set order relations are concerned. We introduce various types of Levitin–Polyak (LP) well-posedness for set optimization problems and survey their relationships. After that, sufficient and necessary conditions for the reference problems to be LP well-posed are given. Furthermore, using the Kuratowski measure of noncompactness, we study characterizations of well-posedness for set optimization problems. Moreover, the links between stability and LP well-posedness of such problems are established via the study on approximating solution mappings. Tools and techniques used in this study and our results are different from existing ones in the literature.

Counter-planning from the kitchen: for a feminist critique of type

Whilst housing has long been a terrain of struggle in terms of its scale, provision, urban morphology and technological advancement, it often escapes a political critique of its interior logic. And yet, it is perhaps only from a political perspective that we might be able to see beyond the impasse we are witnessing. If most of the newly built stock conforms to models established more than a century ago, an increasing number of ‘experimental’ proposals reimagine domesticity with a chequered success that is surprising if we consider how ill-fitting the petit-bourgeois family flat is to our current conditions. In such a conjuncture the concept of type seems to be still a useful ground for debate as it helps us to read housing as a tool for the construction of subjects.At the core of this mandate crisis lies a great unsaid non-said of western society, namely the role played by the house in the institutionalisation of reproductive labour. Reproductive labour is the care, education and actual production of the labour force from childbearing to housework to the care of the elderly—a form of labour that, before mature capitalism, was never seen as separate from other productive activities. In this sense, this paper assumes a feminist standpoint in that it re-reads modern housing as the place of women’s hidden, unwaged work, and typological discourse as the intellectual and technical arsenal that has allowed the fine tuning of such a labour system.The hypothesis that will be explored is that reproductive labour itself is undergoing a large-scale shift that architecture is struggling to register. In order to understand this shift, we will look at the recent architectural production of three nations—the Netherlands, Switzerland and Japan—where a strong design culture has met an acute awareness of the recent changes in the organisation of work. Looking at work by MVRDV, Christian Kerez and SANAA, we will try to construct a map of possible solutions for housing beyond reproductive labour—and, perhaps, beyond type itself.