多凸能量的适位性和正则性

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-01-01 DOI:10.1051/cocv/2023041

Wilfrid Gangbo, Matt Jacobs, Inwon Kim

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引用次数: 0

摘要

我们证明了二维和三维多凸泛函的极小值的存在性、唯一性和正则性，它对应于保测度映射的$H^1$投影。我们的结果基于拉格朗日乘子的小性，引入了最小化器唯一性的新准则。为了得到唯一的全局最小值，不需要对压力的二阶导数进行估计。作为应用，我们构造了一个最小化运动格式来构造短时间区间内Navier-Stokes方程的$L^r$解。

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Well-posedness and regularity for a polyconvex energy

We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the $H^1$ projection of measure-preserving maps. Our result introduces a new criteria on the uniqueness of the minimizer, based on the smallness of the lagrange multiplier. No estimate on the second derivatives of the pressure is needed to get a unique global minimizer. As an application, we construct a minimizing movement scheme to construct $L^r$ solutions of the Navier-Stokes equation for a short time interval.

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来源期刊

Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

自引率

7.10%

发文量

期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.