A Cauchy Problem fo Analytic Functionals.
A direct proof of the equivalence between the entropy solutions of conservation laws and viscosity solutions of Hamilton-Jacobi equations in one-space variable.
A first order partial differential equation with an integral boundary condition
In questo lavoro si considera un’equazione alle derivate parziali del primo ordine con una condizione sulla frontiera di tipo integrale. Si studia resistenza, l'unicità e il comportamento asintotico delle soluzioni.
A mathematical model describing cellular division with a proliferating phase duration depending on the maturity of cells.
A new technique in constructing closed-form solutions for nonlinear PDEs appearing in fluid mechanics and gas dynamics.
A Picard-Maclaurin theorem for initial value PDEs.
An error analysis of the multi-configuration time-dependent Hartree method of quantum dynamics
This paper gives an error analysis of the multi-configuration time-dependent Hartree (MCTDH) method for the approximation of multi-particle time-dependent Schrödinger equations. The MCTDH method approximates the multivariate wave function by a linear combination of products of univariate functions and replaces the high-dimensional linear Schrödinger equation by a coupled system of ordinary differential equations and low-dimensional nonlinear partial differential equations. The main result of this...
An error estimate for the parabolic approximation of multidimensional scalar conservation laws with boundary conditions
An existence theorem for an implicit nonlinear evolution equation.
Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group
We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like where h is the mesh step. Such...
Approximation of solutions of nonlinear initial-value problems by B-spline functions
This note is motivated by [GGG], where an algorithm finding functions close to solutions of a given initial value-problem has been proposed (this algorithm has been recalled in Theorem 2.2). In this paper we present a commonly used definition and basic facts concerning B-spline functions and use them to improve the mentioned algorithm. This leads us to a better estimate of the Cauchy problem solution under some additional assumption on f appearing in the Cauchy problem. We also estimate the accuracy...
Asymptotic solutions for large time of Hamilton–Jacobi equations in euclidean n space
Complex calculus of variations
In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to functions in . It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions...
Continuum limits of particles interacting via diffusion.
Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics
We prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. The equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, which arises in the theory of dislocation dynamics. We show that if an anisotropic mean curvature motion is approximated by equations of this type then it is always of variational type, whereas the converse is true only in dimension two.
Convergence of an accurate scheme for first order quasi linear equations
Difference methods for non-linear partial differential equations of the first order
Estimations hölderiennes pour l'équation d'Euler
Existence and uniqueness of solutions for gradient systems without a compactness embedding condition
This paper is devoted to the existence and uniqueness of solutions for gradient systems of evolution which involve gradients taken with respect to time-variable inner products. The Gelfand triple considered in the setting of this paper is such that the embedding is only continuous.
Existence and Uniqueness of Solutions for Partial Differential-Functional Equations of the First Order with Deviating Argument of the Derivative of Unknown Function
We consider the existence and uniqueness problem for partial differential-functional equations of the first order with the initial condition for which the right-hand side depends on the derivative of unknown function with deviating argument.