Generalized variational solutions of the cauchy problem for hamilton — jacobi equations

1981 ◽  
Vol 6 (3) ◽  
pp. 289-304
Author(s):  
Antonio Leaci ◽  
Giorgio Vergara Caffarelli
Keyword(s):  
Cauchy Problem ◽  
Variational Solutions ◽  
The Cauchy Problem ◽  
Jacobi Equations

scholarly journals VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

2007 ◽  
Vol 04 (04) ◽  
pp. 771-795 ◽  
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.

scholarly journals Cauchy problem and periodic homogenization for nonlocal Hamilton–Jacobi equations with coercive gradient terms

2019 ◽  
Vol 150 (6) ◽  
pp. 3028-3059
Author(s):  
Martino Bardi ◽  
Annalisa Cesaroni ◽  
Erwin Topp
Keyword(s):  
Cauchy Problem ◽  
Uniform Convergence ◽  
Unique Solution ◽  
Comparison Principle ◽  
Nonlocal Operator ◽  
Periodic Homogenization ◽  
Effective Equation ◽  
Superlinear Growth ◽  
The Cauchy Problem ◽  
Jacobi Equations

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.

On Hopf's formula for lipschitz solutions of the cauchy problem for Hamilton-Jacobi equations

1997 ◽  
Vol 29 (10) ◽  
pp. 1145-1159 ◽  
Author(s):  
Tran Duc Van ◽  
Nguyen Hoang ◽  
Mikio Tsuji
Keyword(s):  
Cauchy Problem ◽  
The Cauchy Problem ◽  
Jacobi Equations

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

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