Comparison principle for the Cauchy problem for Hamilton–Jacobi equations with discontinuous data

AbstractThis paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

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On the Cauchy problem associated with the Brinkman flow in

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.47 ◽

2021 ◽

pp. 1-20

Author(s):

Michel Molina Del Sol ◽

Eduardo Arbieto Alarcon ◽

Rafael José Iorio

Keyword(s):

Cauchy Problem ◽

Porous Media ◽

Fluid Flow ◽

Comparison Principle ◽

Well Posedness ◽

Quasilinear Equations ◽

Brinkman Equations ◽

The Cauchy Problem ◽

Model Fluid ◽

Image Position

In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.

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VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891607001355 ◽

2007 ◽

Vol 04 (04) ◽

pp. 771-795 ◽

Cited By ~ 11

Author(s):

GIUSEPPE MARIA COCLITE ◽

NILS HENRIK RISEBRO

Keyword(s):

Cauchy Problem ◽

Viscosity Solution ◽

Viscosity Solutions ◽

Front Tracking ◽

Discontinuous Coefficients ◽

Comparison Result ◽

The Cauchy Problem ◽

Temporal Location ◽

Additional Regularity ◽

Jacobi Equations

We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main result is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to "internal boundaries". The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by "viscosity solution" we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.

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