scholarly journals EIGENELEMENTS OF A GENERAL AGGREGATION-FRAGMENTATION MODEL

2010 ◽  
Vol 20 (05) ◽  
pp. 757-783 ◽  
Author(s):  
MARIE DOUMIC JAUFFRET ◽  
PIERRE GABRIEL
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Asymptotic Behavior Of Solutions ◽  
Fragmentation Model ◽  
Existence Of A Solution ◽  
Priori Estimates ◽  
Long Time ◽  
Weighted Norms ◽  
Fragmentation Processes ◽  
Time Asymptotic Behavior

We consider a linear integro-differential equation which arises to describe both aggregation-fragmentation processes and cell division. We prove the existence of a solution (λ, [Formula: see text], ϕ) to the related eigenproblem. Such eigenelements are useful to study the long-time asymptotic behavior of solutions as well as the steady states when the equation is coupled with an ODE. Our study concerns a non-constant transport term that can vanish at x = 0, since it seems to be relevant to describe some biological processes like proteins aggregation. Non-lower-bounded transport terms bring difficulties to find a priori estimates. All the work of this paper is to solve this problem using weighted-norms.

EXISTENCE OF A SOLUTION TO THE CELL DIVISION EIGENPROBLEM

2006 ◽  
Vol 16 (supp01) ◽  
pp. 1125-1153 ◽  
Author(s):  
PHILIPPE MICHEL
Keyword(s):  
Cell Division ◽  
A Priori Estimates ◽  
A Priori ◽  
Existence Of A Solution ◽  
Continuous Growth ◽  
Time Asymptotics ◽  
Priori Estimates ◽  
Long Time ◽  
Long Time Asymptotics ◽  
Number Of Cells

We consider the cell division equation which describes the continuous growth of cells and their division in two pieces. Growth conserves the total number of cells while division conserves the total mass of the system but increases the number of cells. We give general assumptions on the coefficient so that we can prove the existence of a solution (λ, N, ϕ) to the related eigenproblem. We also prove that the solution can be obtained as the sum of an explicit series. Our motivation, besides its applications to the biology and fragmentation, is that the eigenelements allow to prove a priori estimates and long-time asymptotics through the General Relative Entropy.16.

LIFESPAN ESTIMATES FOR SOLUTIONS OF POLYHARMONIC KIRCHHOFF SYSTEMS

2012 ◽  
Vol 22 (02) ◽  
pp. 1150009 ◽  
Author(s):  
GIUSEPPINA AUTUORI ◽  
FRANCESCA COLASUONNO ◽  
PATRIZIA PUCCI
Keyword(s):  
A Priori Estimates ◽  
Driving Forces ◽  
A Priori ◽  
Thin Plates ◽  
Existence Of A Solution ◽  
Dissipative Term ◽  
Maximal Solutions ◽  
Elastic Bodies ◽  
Priori Estimates ◽  
Deformation Processes

In mathematical physics we increasingly encounter PDEs models connected with vibration problems for elastic bodies and deformation processes, as it happens in the Kirchhoff–Love theory for thin plates subjected to forces and moments. Recently Monneanu proved in Refs. 26 and 27 the existence of a solution of the nonlinear Kirchhoff–Love plate model. In this paper we treat several questions about non-continuation for maximal solutions of polyharmonic Kirchhoff systems, governed by time-dependent nonlinear dissipative and driving forces. In particular, we are interested in the strongly damped Kirchhoff–Love model, containing also an intrinsic dissipative term of Kelvin–Voigt type. Global non-existence and a priori estimates for the lifespan of maximal solutions are proved. Several applications are also presented in special subcases of the source term f and the nonlinear external damping Q.

A nonlinear partial differential system arising in thermoelectricity

2005 ◽  
Vol 16 (6) ◽  
pp. 683-712 ◽  
Author(s):  
A. BERMÚDEZ ◽  
R. MUÑOZ-SOLA ◽  
F. PENA
Keyword(s):  
Source Term ◽  
A Priori Estimates ◽  
A Priori ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Existence Of A Solution ◽  
Nonlinear Parabolic ◽  
Priori Estimates ◽  
Partial Differential System ◽  
Temperature Equation

In this paper we prove the existence of a solution for a system of nonlinear parabolic partial differential equations arising from thermoelectric modelling of metallurgical electrodes undergoing a phase change. The model consists of an electromagnetic problem for eddy current computation coupled with a Stefan problem for temperature. The proof uses a regularized problem obtained by truncating the source term in temperature equation. Passing to the limit requires fine a priori estimates leading to compactness.

scholarly journals On travelling wave solutions of a model of a liquid film flowing down a fibre

2021 ◽  
pp. 1-30
Author(s):  
HANGJIE JI ◽  
ROMAN TARANETS ◽  
MARINA CHUGUNOVA
Keyword(s):  
Thin Film ◽  
Weak Solutions ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Travelling Wave ◽  
Model Parameters ◽  
Priori Estimates ◽  
Thin Film Model ◽  
Long Time

Abstract Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

An inverse problem for the KdV equation with Neumann boundary measured data

2018 ◽  
Vol 26 (1) ◽  
pp. 133-151 ◽  
Author(s):  
Sakthivel Kumarasamy ◽  
Alemdar Hasanov
Keyword(s):  
Inverse Problem ◽  
A Priori Estimates ◽  
Kdv Equation ◽  
A Priori ◽  
Gradient Methods ◽  
Neumann Boundary ◽  
Sobolev Norm ◽  
Minimum Problem ◽  
Existence Of A Solution ◽  
Priori Estimates

AbstractIn this paper, we consider an inverse coefficient problem for the linearized Korteweg–de Vries (KdV) equation {u_{t}+u_{xxx}+(c(x)u)_{x}=0}, with homogeneous boundary conditions {u(0,t)=u(1,t)=u_{x}(1,t)=0}, when the Neumann data{g(t):=u_{x}(0,t)}, {t\in(0,T)}, is given as an available measured output at the boundary {x=0}. The inverse problem is formulated as a minimum problem for the regularized Tikhonov functional {\mathcal{J}_{\alpha}(c)=\frac{1}{2}\|u_{x}(0,\cdot\,;c)-g\|^{2}_{L^{2}(0,T)}+% \frac{\alpha}{2}\|c^{\prime}\|^{2}_{L^{2}(0,1)}} with Sobolev norm. Based on a priori estimates for the weak and regular weak solutions of the direct and adjoint problems, it is proved that the input-output operator is compact, which shows the ill-posedness of the inverse problem. Then Fréchet differentiability of the Tikhonov functional and Lipschitz continuity of the Fréchet gradient are proved. It is shown that the last result allows us to use an important advantage of gradient methods when the functional is from the class {C^{1,1}(\mathcal{M})}. In the final part, an existence of a solution of the minimum problem for the regularized Tikhonov functional {\mathcal{J}_{\alpha}(c)} is proved.

Three-Level Difference Schemes on Non-Uniform in Time Grids

2001 ◽  
Vol 1 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Piotr Matus ◽  
Elena Zyuzina
Keyword(s):  
Hilbert Spaces ◽  
Wave Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Approximation ◽  
Difference Schemes ◽  
Nonuniform Grids ◽  
Priori Estimates ◽  
Long Time ◽  
The Right

Abstract In this work, a stability of three-level operator-difference schemes on nonuniform in time grids in Hilbert spaces is studied. A priori estimates of a long time stability (for t → ∞) in the sense of the initial data and the right-hand side are obtained in different energy norms without demanding the quasiuniformity of the grid. New difference schemes of the second order of local approximation on nonuniform grids both in time and space on standard stencils for parabolic and wave equations are adduced.

EXISTENCE OF SOLUTIONS TO A THERMO-VISCOELASTIC CONTACT PROBLEM WITH COULOMB FRICTION

2002 ◽  
Vol 12 (10) ◽  
pp. 1491-1511 ◽  
Author(s):  
C. ECK
Keyword(s):  
Contact Problem ◽  
Coulomb Friction ◽  
A Priori Estimates ◽  
A Priori ◽  
Regularity Result ◽  
Existence Of A Solution ◽  
Moderate Growth ◽  
Priori Estimates ◽  
The Galerkin Method ◽  
Viscoelastic Contact

A thermo-viscoelastic contact problem with Coulomb friction is studied. The model involves a temperature-dependent heat conductivity with moderate growth in order to control the rapidly growing mixed terms as e.g. the heat generated by friction or by viscous deformation. The contact condition is formulated in velocities. The existence of a solution is proved by an approximation of the problem in several steps involving a penalty-approximation and a smoothing of the friction law. The solvability of the approximate problem is proved by the Galerkin method. A priori estimates uniform with respect to all approximation parameters make it possible to pass to the original problem. These estimates are based on a regularity result for the contact problem without heat transfer.

scholarly journals Solvability of a multi-point boundary value problem of Neumann type

1999 ◽  
Vol 4 (2) ◽  
pp. 71-81 ◽  
Author(s):  
Chaitan P. Gupta ◽  
Sergei Trofimchuk
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Type A ◽  
Point Boundary ◽  
Existence Of A Solution ◽  
Priori Estimates ◽  
Neumann Type ◽  
Existence Conditions

Letf:[0,1]×ℝ2→ℝbe a function satisfying Carathéodory's conditions ande(t)∈L1[0,1]. Letξi∈(0,1),ai∈ℝ,i=1,2,…,m−2,0<ξ1<ξ2<⋯<ξm−2<1be given. This paper is concerned with the problem of existence of a solution for them-point boundary value problemx″(t)=f(t,x(t),x′(t))+e(t),0<t<1;x(0)=0,x′(1)=∑i=1m−2ai x′(ξi). This paper gives conditions for the existence of a solution for this boundary value problem using some new Poincaré type a priori estimates. This problem was studied earlier by Gupta, Ntouyas, and Tsamatos (1994) when all of theai∈ℝ,i=1,2,…,m−2, had the same sign. The results of this paper give considerably better existence conditions even in the case when all of theai∈ℝ,i=1,2,…,m−2, have the same sign. Some examples are given to illustrate this point.

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