Search results for "Weak formulation"
showing 6 items of 6 documents
Wave propagation in 1D elastic solids in presence of long-range central interactions
2011
Abstract In this paper wave propagation in non-local elastic solids is examined in the framework of the mechanically based non-local elasticity theory established by the author in previous papers. It is shown that such a model coincides with the well-known Kroner–Eringen integral model of non-local elasticity in unbounded domains. The appeal of the proposed model is that the mechanical boundary conditions may easily be imposed because the applied pressure at the boundaries of the solid must be equilibrated by the Cauchy stress. In fact, the long-range forces between different volume elements are modelled, in the body domain, as central body forces applied to the interacting elements. It is …
Renormalized solutions for degenerate elliptic–parabolic problems with nonlinear dynamical boundary conditions and L1-data
2008
Abstract We consider a degenerate elliptic–parabolic problem with nonlinear dynamical boundary conditions. Assuming L 1 -data, we prove existence and uniqueness in the framework of renormalized solutions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problems and in the weak formulation of the mathematical model of the so-called Hele–Shaw problem. Also, the problem with non-homogeneous Neumann boundary condition is included.
Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions
2006
We study the questions of existence and uniqueness of weak and entropy solutions for equations of type -div a(x, Du)+γ(u) ∋ φ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x, Du)·η+β(u) ∋ ψ. The nonlinear elliptic operator div a(x, Du) is modeled on the p-Laplacian operator Δp(u) = div (|Du|p−2Du), with p > 1, γ and β are maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) and 0 ∈ β(0), and the data φ ∈ L1 (Ω) and ψ ∈ L1 (∂Ω). We also study existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with c…
Theoretical study of a Bénard Marangoni problem
2011
[EN] In this paper we prove the existence of strong solutions for the stationary Benard-Marangoni problem in a finite domain flat on the top, bifurcating from the basic heat conductive state. The Benard-Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled with a system of partial differential equations of the type Navier-Stokes and heat equation. The boundary conditions include crossed boundary conditions involving tangential derivatives of the temperature and normal derivatives of the velocity field. To define tangential derivatives at the boundary, intended in the trace sense,…
Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term
2001
Abstract Our aim in this article is to study the following nonlinear elliptic Dirichlet problem: − div [a(x,u)·∇u]+b(x,u,∇u)=f, in Ω; u=0, on ∂Ω; where Ω is a bounded open subset of RN, with N>2, f∈L m (Ω) . Under wide conditions on functions a and b, we prove that there exists a type of solution for this problem; this is a bounded weak solution for m>N/2, and an unbounded entropy solution for N/2>m⩾2N/(N+2). Moreover, we show when this entropy solution is a weak one and when can be taken as test function in the weak formulation. We also study the summability of the solutions.
A Wavelet-Galerkin Method for a 1D Elastic Continuum with Long- Range Interactions
2009
An elastic continuum model with long-range forces is addressed in this study. The model stems from a physically-based approach to non-local mechanics where non-adjacent volume elements exchange mutual central forces that depend on the relative displacement and on the product between the interacting volume elements; further, they are taken as proportional to a material dependent and distance-decaying function. Smooth-decay functions lead to integrodifferential equations while hypersingular, fractional-decay functions lead to a fractional differential equation of Marchaud type. In both cases the governing equations are solved by the Galerkin method with different sets of basis functions, amon…