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Solutions of a nonhyperbolic pair of balance laws

Michael Sever (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We describe a constructive algorithm for obtaining smooth solutions of a nonlinear, nonhyperbolic pair of balance laws modeling incompressible two-phase flow in one space dimension and time. Solutions are found as stationary solutions of a related hyperbolic system, based on the introduction of an artificial time variable. As may be expected for such nonhyperbolic systems, in general the solutions obtained do not satisfy both components of the given initial data. This deficiency may be overcome,...

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

We study the existence of positive solutions to ⎧ d i v ( | ∇ u | p - 2 ∇ u ) + q ( x ) u - γ = 0 on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is ℝ N or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

Solutions to a perturbed critical semilinear equation concerning the N -Laplacian in ℝ N

Elliot Tonkes (1999)

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to study the existence of variational solutions to a nonhomogeneous elliptic equation involving the N -Laplacian - Δ N u ≡ - div ( | ∇ u | N - 2 ∇ u ) = e ( x , u ) + h ( x ) in Ω where u ∈ W 0 1 , N ( ℝ N ) , Ω is a bounded smooth domain in ℝ N , N ≥ 2 , e ( x , u ) is a critical nonlinearity in the sense of the Trudinger-Moser inequality and h ( x ) ∈ ( W 0 1 , N ) * is a small perturbation.

Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

Leonid Berlyand, Volodymyr Rybalko (2010)

Journal of the European Mathematical Society

We study solutions of the 2D Ginzburg–Landau equation - Δ u + ε - 2 u ( | u | 2 - 1 ) = 0 subject to “semi-stiff” boundary conditions: Dirichlet conditions for the modulus, | u | = 1 , and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small ε . For the Dirichlet boundary condition (“stiff” problem), the existence of stable solutions with vortices, whose energy...

Soluzioni periodiche di PDEs Hamiltoniane

Massimiliano Berti (2004)

Bollettino dell'Unione Matematica Italiana

Presentiamo nuovi risultati di esistenza e molteplicità di soluzioni periodiche di piccola ampiezza per equazioni alle derivate parziali Hamiltoniane. Otteniamo soluzioni periodiche di equazioni «completamente risonanti» aventi nonlinearità generali grazie ad una riduzione di tipo Lyapunov-Schmidt variazionale ed usando argomenti di min-max. Per equazioni «non risonanti» dimostriamo l'esistenza di soluzioni periodiche di tipo Birkhoff-Lewis, mediante un'opportuna forma normale di Birkhoff e realizzando...

Solvability of a class of phase field systems related to a sliding mode control problem

Michele Colturato (2016)

Applications of Mathematics

We consider a phase-field system of Caginalp type perturbed by the presence of an additional maximal monotone nonlinearity. Such a system arises from a recent study of a sliding mode control problem. We prove the existence of strong solutions. Moreover, under further assumptions, we show the continuous dependence on the initial data and the uniqueness of the solution.

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