1,720,983 research outputs found
A note on positive solutions for conservation law with singular source
Full text linkWe consider the Cauchy problem for the scalar conservation law
∂tu+∂xf(u)= 1/g(u), t>0, x∈R,
with g ∈ C^1(R), g(0) = 0, g(u) > 0 for u > 0, and assume that the initial datum u0 is nonnegative. We show the existence of entropy solutions that are positive a.e., by means of an approximation of the equation that preserves positive solutions, and by passing to the limit using a monotonicity argument. The difficulty lies in handling the singularity of the right hand side (the source term) as
u possibly vanishes at the initial time. The source term is shown to be locally integrable. Moreover, we prove an uniqueness and stability result for the above equation
On a singular limit as for a model for the evolution of morphogens in a growing tissue
No full textThis paper is devoted to the singular limit of a model for the regulation of growth and patterning in developing tissues by diffusing morphogens. The model is governed by a system of nonlinear PDEs. The arguments are based on energy estimates and a change of variable that reduces the system into a nonlinear PDE with singular diffusion
Long time behavior of a model for the evolution of morphogens in a growing tissue II: θ
Full text linkWe consider a model for the regulation of growth and patterning in developing tissues by diffusing morphogens expressed in terms of a system of nonlinear PDEs. Transforming such a system in an equation with singular diffusion we analyze its long time behavior
Regularity and energy transfer for a nonlinear beam equation
No full textIn this paper we study some key effects of a discontinuous forcing term in a fourth order wave equation on a bounded domain, modeling the adhesion of an elastic beam with a substrate through an elastic-breakable interaction. By using a spectral decomposition method we show that the main effects induced by the nonlinearity at the transition from attached to detached states can be traced in a loss of regularity of the solution and in a migration of the total energy through the scales
A nonlocal Lagrangian traffic flow model and the zero-filter limit
Full text linkIn this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the L sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy
Dispersive Effects in Two- and Three-Dimensional Peridynamics
No full textIn this paper we study the dispersive properties related to a model of peridynamic evolution, governed by a non local initial value problem, in the cases of two and three spatial dimensions. The features of the wave propagation characterized by the nontrivial interactions between nonlocality and the regimes of low and high frequencies are studied and suitable numerical investigations are exposed
Some observations regarding the stationary Buckley–Leverett equation
No full textThe basic hyperbolic–elliptic black-oil model describes oil–water displacement in a porous
medium. Given its mathematical complexity, there is a need for particular simple solutions
for validation of numerical methods. We present a class of stationary solutions, which are easy
to compute, and in many cases are given by explicit formulae. These solutions are constructed
by a nonlinear coupling of two linear equations, an elliptic pressure equation and a hyperbolic
saturation equation
Adhesion and debonding in a model of elastic string
Full text linkWe study the problem of adhesion and debonding of an elastic body interacting with a rigid substrate through a layer made of soft breakable adhesive material. The general problem is formulated in the multidimensional vectorial case, while a detailed analysis is carried for the one dimensional case leading to the study of a 1D semilinear wave equation. The initial boundary value problem is affected by the presence of a source term characterizing the interaction of the string withe the adhesive layer. This discontinuous force jumps to zero when a critical value of the displacement is reached. We obtain conditions for the initial boundary value problem ensuring the regularity of the solutions and the attachment-detachment conditions. Finally, we focus on a numerical investigation of the problem by considering regularizations of the source term and different initial conditions. (C) 2019 Elsevier Ltd. All rights reserved
Vanishing viscosity versus Rosenau approximation for scalar conservation laws: The fractional case
No full textWe consider approximations of scalar conservation laws by adding nonlocal diffusive operators. In particular,
we consider solutions associated to fractional Laplacian and fractional Rosenau perturbations and
show that, for any t > 0, the mutual L1 distance of their profiles is negligible as compared to their common
distance to the underlying inviscid entropy solution. We provide explicit examples showing that our rates
are optimal in the supercritical and critical cases, in one space dimension and for the Burgers’ flux. For
subcritical equations, our rates are not optimal but they remain explicit
Well-posedness of the Initial Value Problem for the Ostrovsky–Hunter Equation with Spatially Dependent Flux
No full textIn this paper we study the Ostrovsky-Hunter equation for the case where the flux function f(x, u) may depend on the spatial variable with certain smoothness. Our main results are that if the flux function is smooth enough (namely f(x)(x, u) is uniformly Lipschitz locally in u and f(u)(x, u) is uniformly bounded), then there exists a unique entropy solution. To show the existence, after proving some a priori estimates we have used the method of compensated compactness and to prove the uniqueness we have employed the method of doubling of variables
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