Recent zbMATH articles in MSC 45Khttps://zbmath.org/atom/cc/45K2021-06-15T18:09:00+00:00WerkzeugConvergence of some mean field games systems to aggregation and flocking models.https://zbmath.org/1460.353492021-06-15T18:09:00+00:00"Bardi, Martino"https://zbmath.org/authors/?q=ai:bardi.martino"Cardaliaguet, Pierre"https://zbmath.org/authors/?q=ai:cardaliaguet.pierreThe authors investigate the convergence of solutions for two classes of Mean Field Game (MFG) systems. The first class of MFG systems with control on the velocity is given by a parabolic system with a large parameter \(\lambda\) associated to a stochastic MFG, for which the solution converges to a solution of a aggregation model as \(\lambda\to\infty\). The second class of MFG systems with control on acceleration is a first order PDEs system for which the solution converges to the solution of a kinetic equation. In the main results, they use PDEs methods for the first model, and variational methods in the space of probability measures on trajectories for the second model.
Reviewer: Rodica Luca (Iaşi)Well-posedness of abstract integro-differential equations with state-dependent delay.https://zbmath.org/1460.340922021-06-15T18:09:00+00:00"Hernández, Eduardo"https://zbmath.org/authors/?q=ai:hernandez.eduardo-m"Fernandes, Denis"https://zbmath.org/authors/?q=ai:fernandes.denis"Wu, Jianhong"https://zbmath.org/authors/?q=ai:wu.jianhong.1|wu.jianhongThe aim of this paper is to study the well-posedness, existence and uniqueness of strict solutions for the following general class of abstract integro-differential equations with state-dependent delay
\[
\left\{\begin{aligned}
u'(t)&=Au(t)+F(t,u(t),
\int_{0}^{t}K(t,\tau)u(\tau-\sigma(\tau,u(\tau)))~d\tau),\quad\text{for}\quad t\in[0,a], 0\\
u_{0}&=\varphi \in C_{Lip}\left( \left[ -p,0\right] ;X\right) ,
\end{aligned}\right.
\]
where, \(A\) is the infinitesimal generator of an analytic semigroup of bounded linear operators on a Banach space \((X,\left\| .\right\| ),\) \(K(.)\) is an operator of valued map which satisfies the natural and nonrestrictive integrability condition, and \(F, \sigma\) are continuous functions.
Motivated by some of their previous works, the authors obtain, under certain appropriate conditions, some useful and fundamental estimates that are essential to proving the main theoretical results. In order to illustrate the feasibility and effectiveness of these results, the authors present some important examples.
Compared with the existing results in the literatures, the results of this study are more general.
Reviewer: Mohamed Zitane (Meknès)Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations.https://zbmath.org/1460.651602021-06-15T18:09:00+00:00"Bailo, Rafael"https://zbmath.org/authors/?q=ai:bailo.rafael"Carrillo, José A."https://zbmath.org/authors/?q=ai:carrillo.jose-antonio"Murakawa, Hideki"https://zbmath.org/authors/?q=ai:murakawa.hideki"Schmidtchen, Markus"https://zbmath.org/authors/?q=ai:schmidtchen.markusSummary: We study an implicit finite-volume scheme for nonlinear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in our work [Commun. Math. Sci. 18, No. 5, 1259--1303 (2020; Zbl 07342310)]. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.
Reviewer: Reviewer (Berlin)Invariant manifolds of a weakly dissipative version of the nonlocal Ginzburg-Landau equation.https://zbmath.org/1460.930482021-06-15T18:09:00+00:00"Kulikov, A. N."https://zbmath.org/authors/?q=ai:kulikov.a-n"Kulikov, D. A."https://zbmath.org/authors/?q=ai:kulikov.dmitrii-anatolevichSummary: We consider a periodic boundary value problem for a nonlocal Ginzburg-Landau equation in its weakly dissipative version. The existence, stability, and local bifurcations of one-mode periodic solutions are studied. It is shown that in a neighborhood of one-mode periodic solutions there may exist a three-dimensional local attractor filled with spatially inhomogeneous time-periodic solutions. Asymptotic formulas for these solutions are obtained. The results are based on using and developing methods of the theory of infinite-dimensional dynamical systems. In a special version of the partial integro-differential equation considered, we study the existence of a global attractor. Solution in the form of series are obtained for this version of the nonlinear boundary value problem.
Reviewer: Reviewer (Berlin)A mixed discrete-continuous fragmentation model.https://zbmath.org/1460.470212021-06-15T18:09:00+00:00"Baird, Graham"https://zbmath.org/authors/?q=ai:baird.graham"Süli, Endre"https://zbmath.org/authors/?q=ai:suli.endre-eSummary: Motivated by the occurrence of ``shattering'' mass-loss observed in purely continuous fragmentation models, this work concerns the development and the mathematical analysis of a new class of hybrid discrete-continuous fragmentation models. Once established, the model, which takes the form of an integro-differential equation coupled with a system of ordinary differential equations, is subjected to a rigorous mathematical analysis, using the theory and methods of operator semigroups and their generators. Most notably, by applying the theory relating to the Kato-Voigt perturbation theorem, honest substochastic semigroups and operator matrices, the existence of a unique, differentiable solution to the model is established. This solution is also shown to preserve nonnegativity and conserve mass.
Reviewer: Reviewer (Berlin)On a discrete scheme for time fractional fully nonlinear evolution equations.https://zbmath.org/1460.353632021-06-15T18:09:00+00:00"Giga, Yoshikazu"https://zbmath.org/authors/?q=ai:giga.yoshikazu"Liu, Qing"https://zbmath.org/authors/?q=ai:liu.qing.1"Mitake, Hiroyoshi"https://zbmath.org/authors/?q=ai:mitake.hiroyoshiSummary: We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo's time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.
Reviewer: Reviewer (Berlin)Numerical solution of the neural field equation in the presence of random disturbance.https://zbmath.org/1460.650082021-06-15T18:09:00+00:00"Kulikov, G. Yu."https://zbmath.org/authors/?q=ai:kulikov.gennady-yurevich"Lima, Pedro M."https://zbmath.org/authors/?q=ai:lima.pedro-miguel"Kulikova, Maria V."https://zbmath.org/authors/?q=ai:kulikova.maria-vSummary: This paper aims at presenting an efficient and accurate numerical method for treating both deterministic- and stochastic-type \textit{neural field equations} (NFEs) in the presence of external stimuli input (or without it). The devised numerical integration means belongs to the class of Galerkin-type spectral approximations. The particular effort is focused on an efficient practical implementation of the solution technique because of the partial integro-differential fashion of the NFEs in use, which are to be integrated, numerically. Our method is implemented in Matlab. Its practical performance and efficiency are investigated on three variants of an NFE model with external stimuli inputs. We study both the deterministic case of the mentioned model and its stochastic counterpart to observe important differences in the solution behavior. First, we observe only stable one-bump solutions in the deterministic neural field scenario, which, in general, will be preserved in our stochastic NFE scenario if the level of random disturbance is sufficiently small. Second, if the area of the external stimuli is large enough and exceeds the size of the bump, considerably, the stochastic neural field solution's behavior may change dramatically and expose also two- and three-bump patterns. In addition, we show that strong random disturbances, which may occur in neural fields, fully alter the behavior of the deterministic NFE solution and allow for multi-bump (and even periodic-type) solutions to appear in all variants of the stochastic NFE model studied in this paper.
Reviewer: Reviewer (Berlin)Nonlocal and nonlinear evolution equations in perforated domains.https://zbmath.org/1460.450042021-06-15T18:09:00+00:00"Pereira, Marcone C."https://zbmath.org/authors/?q=ai:pereira.marcone-correa"Sastre-Gomez, Silvia"https://zbmath.org/authors/?q=ai:sastre-gomez.silviaThe authors describe the behavior when \(\varepsilon \rightarrow 0\) of the solution to a nonlocal evolution equation written as
\[
u_{t}(x,t)=\int_{ \mathbb{R}^{N}\backslash (\Omega \backslash \Omega ^{\varepsilon })}J(x-y)u(y,t)dy-h_{\varepsilon }(x)u(x,t)+f(x,u(x,t)),
\]
with \((x,t)\in \Omega ^{\varepsilon }\times (0,\infty )\), where \(\Omega ^{\varepsilon }\) is included in a fixed and bounded domain \(\Omega \subset \mathbb{R}^{N}\) and is such that its characteristic function \(\chi _{\varepsilon }\) weakly\( ^{\ast }\) converges in \(L^{\infty }(\Omega )\) to some function \(\chi \in L^{\infty }(\mathbb{R}^{N})\) which is positive in \(\Omega \). With this hypothesis, the authors cover more situations than that of a perforated domain. Dirichlet or Neumann boundary conditions are added, together with the initial data \(u(x,0)=u_{0}(x)\). In the above equation, \(J\in C(\mathbb{R} ^{N},\mathbb{R})\) is a smooth, non-negative and non-singular kernel which satisfies \(J(0)>0\), \(J(-x)=J(x)\) and \(\int_{\mathbb{R}^{N}}J(x)dx=1\). The function \(f:\Omega ^{\varepsilon }\times L^{1}(\Omega ^{\varepsilon })\rightarrow \mathbb{R}\) is defined through \(f(x,u)=(g\circ m_{\Omega ^{\varepsilon }})(x,u)\), where \(g:\mathbb{R}\rightarrow \mathbb{R}\) is a smooth and globally Lipschitz function, and \(m_{\Omega ^{\varepsilon }}:\Omega ^{\varepsilon }\times L^{1}(\Omega ^{\varepsilon })\rightarrow \mathbb{R}\) is defined as \(m_{\Omega ^{\varepsilon }}=\frac{1}{\left\vert B_{\delta }(x)\cap \Omega ^{\varepsilon }\right\vert }\int_{B_{\delta }(x)\cap \Omega ^{\varepsilon }}u(y)dy\).
The first main result deals with the case where \(h_{\varepsilon }=1\) and proves the existence of a unique limit \(u^{\ast }:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}\) with \(u^{\ast }(x,t)\equiv 0\) in \(\mathbb{R}^{N}\backslash \Omega \) and \(u^{\ast }\in C^{1}([a,b],L^{2}(\mathbb{R}^{N}))\) for any closed interval \( [a,b]\subset \mathbb{R}^{N}\), such that \(u_{\varepsilon }\rightarrow u^{\ast }\) weakly\(^{\ast}\) in \(L^{\infty }([a,b];L^{2}(\Omega ))\), as \(\varepsilon \rightarrow 0\). Moreover, this limit \(u^{\ast }\) is the solution to the nonlocal equation
\[
u_{t}(x,t)=\chi (x)\int_{\mathbb{R} ^{N}}J(x-y)(u(y,t)-u(x,t))dy+\chi (x)f_{\chi }(x,u(x,t))+(\chi (x)-1)u(x,t),
\]
in \(\Omega \times (0,\infty )\), with the initial condition \(u(x,0)=\chi (x)u_{0}(x)\). Here \(f_{\chi }(x,u)=(g\circ m_{\chi })(x,u)\), with \(m_{\chi }(x,u)=\frac{1}{\int_{B_{\delta }(x)}\chi (y)dy}\int_{B_{\delta }(x)}u(y)dy\).
The second main result deals with the case where
\[
h_{\varepsilon }(x)=\int_{\mathbb{R}^{N}\backslash (\Omega \backslash \Omega ^{\varepsilon })}J(x-y)dy \text{ for } x\in \mathbb{R}^{N}.
\]
The authors here prove a similar asymptotic behavior, but with a slightly different limit nonlocal equation. The limit solutions depend on \(\chi \) and on the small parameter \(\delta \). The third main result of the paper describes the asymptotic behavior of these limit solutions as \(\delta \rightarrow 0\), under further hypotheses on the data.
For the proofs of these results, the authors first prove the existence of a unique solution \(u^{\varepsilon }\in C^{1}([a,b];L^{2}(\Omega ^{\varepsilon }))\) to the original problem on which they establish uniform estimates, considering the first eigenvalue of the problem. They also establish properties of the limit problem for which they prove the existence of a unique solution. The convergence results are obtained through direct computations and using the dominated convergence theorem.
Reviewer: Alain Brillard (Riedisheim)