Approximate Solutions and α-Well-Posedness for Variational Inequalities and Nash Equilibria

This paper is devoted to the mathematical modeling and analysis of a hyperbolic Maxwell quasi-variational inequality (QVI) for the Bean-Kim superconductivity model with temperature and magnetic field dependence in the critical current. Emerging from the Euler time discretization, we analyze the corresponding H(curl)-elliptic QVI and prove its existence using a fixed-point argument in combination with techniques from variational inequalities and Maxwell's equations. Based on the existence result for the H(curl)-elliptic QVI, we examine the stability and convergence of the Euler scheme, which serve as our fundament for the well-posedness of the governing hyperbolic Maxwell QVI.

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Hyperbolic Maxwell variational inequalities of the second kind

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2019015 ◽

2020 ◽

Vol 26 ◽

pp. 34 ◽

Cited By ~ 1

Author(s):

Irwin Yousept

Keyword(s):

Variational Inequalities ◽

Existence Result ◽

Semigroup Theory ◽

Regularity Property ◽

Well Posedness ◽

Test Functions ◽

Local Boundedness ◽

Subdifferential Calculus ◽

Faraday’S Law ◽

Yosida Regularization

We analyze a class of hyperbolic Maxwell variational inequalities of the second kind. By means of a local boundedness assumption on the subdifferential of the underlying nonlinearity, we prove a well-posedness result, where the main tools for the proof are the semigroup theory for Maxwell’s equations, the Yosida regularization and the subdifferential calculus. The second part of the paper focuses on a more general case omitting the local boundedness assumption. In this case, taking into account more regular initial data and test functions, we are able to prove a weaker existence result through the use of the minimal section operator associated with the Nemytskii operator of the governing subdifferential. Eventually, we transfer the developed well-posedness results to the case involving Faraday’s law, which in particular allows us to improve the regularity property of the electric field in the weak existence result.

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Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

Computational Management Science ◽

10.1007/s10287-009-0093-8 ◽

2009 ◽

Vol 6 (3) ◽

pp. 373-375 ◽

Cited By ~ 18

Author(s):

Jong-Shi Pang ◽

Masao Fukushima

Keyword(s):

Variational Inequalities ◽

Nash Equilibria ◽

Generalized Nash Equilibria

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Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

International Mathematics Research Notices ◽

10.1093/imrn/rnz385 ◽

2020 ◽

Author(s):

Bjoern Bringmann

Keyword(s):

Wave Equation ◽

Nonlinear Wave ◽

Nonlinear Wave Equation ◽

Local Existence ◽

Approximate Solutions ◽

Well Posedness ◽

Dispersive Equations ◽

Deterministic Theory ◽

Nonlinear Smoothing ◽

Iterative Decomposition

Abstract We study the derivative nonlinear wave equation $- \partial _{tt} u + \Delta u = |\nabla u|^2$ on $\mathbb{R}^{1 +3}$. The deterministic theory is determined by the Lorentz-critical regularity $s_L = 2$, and both local well-posedness above $s_L$ as well as ill-posedness below $s_L$ are known. In this paper, we show the local existence of solutions for randomized initial data at the super-critical regularities $s\geqslant 1.984$. In comparison to the previous literature in random dispersive equations, the main difficulty is the absence of a (probabilistic) nonlinear smoothing effect. To overcome this, we introduce an adaptive and iterative decomposition of approximate solutions into rough and smooth components. In addition, our argument relies on refined Strichartz estimates, a paraproduct decomposition, and the truncation method of de Bouard and Debussche.

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