412 research outputs found
PDEs in Moving Time Dependent Domains
In this work we study partial differential equations defined in a domain that
moves in time according to the flow of a given ordinary differential equation,
starting out of a given initial domain. We first derive a formulation for a
particular case of partial differential equations known as balance equations.
For this kind of equations we find the equivalent partial differential
equations in the initial domain and later we study some particular cases with
and without diffusion. We also analyze general second order differential
equations, not necessarily of balance type. The equations without diffusion are
solved using the characteristics method. We also prove that the diffusion
equations, endowed with Dirichlet boundary conditions and initial data, are
well posed in the moving domain. For this we show that the principal part of
the equivalent equation in the initial domain is uniformly elliptic. We then
prove a version of the weak maximum principle for an equation in a moving
domain. Finally we perform suitable energy estimates in the moving domain and
give sufficient conditions for the solution to converge to zero as time goes to
infinity.Comment: pp 559-577. Without Bounds: A Scientific Canvas of Nonlinearity and
Complex Dynamics (2013) p. 36
Verifications of primal energy identities for variational problems with obstacles
We discuss error identities for two classes of free boundary problems
generated by obstacles. The identities suggest true forms of the respective
error measures which consist of two parts: standard energy norm and a certain
nonlinear measure. The latter measure controls (in a weak sense) approximation
of free boundaries. Numerical tests confirm sharpness of error identities and
show that in different examples one or another part of the error measure may be
dominant.Comment: 8 pages, 2 figures, conference paper: LSSC (Large-Scale scientific
computing), Sozopol, Bulgaria, 2017. The final version will be published at
Springe
Homogénéisation par éléments finis des modules de comportements élastiques de matériaux composites
Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics
We consider a coupled bulk-surface system of partial differential equations with nonlinear coupling modelling receptor-ligand dynamics. The model arises as a simpliïŹcation of a mathematical model for the reaction between cell surface resident receptors and ligands present in the extra-cellular medium. We prove the existence and uniqueness of solutions. We also consider a number of biologically relevant asymptotic limits of the model. We prove convergence to limiting problems which take the form of free boundary problems posed on the cell surface. We also report on numerical simulations illustrating convergence to one of the limiting problems as well as the spatio-temporal distributions of the receptors and ligands in a realistic geometry
Quasivariational solutions for first order quasilinear equations with gradient constraint
We prove the existence of solutions for an evolution quasi-variational
inequality with a first order quasilinear operator and a variable convex set,
which is characterized by a constraint on the absolute value of the gradient
that depends on the solution itself. The only required assumption on the
nonlinearity of this constraint is its continuity and positivity. The method
relies on an appropriate parabolic regularization and suitable {\em a priori}
estimates. We obtain also the existence of stationary solutions, by studying
the asymptotic behaviour in time. In the variational case, corresponding to a
constraint independent of the solution, we also give uniqueness results
Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system
This paper deals with the derivation and analysis of the the Hall
Magneto-Hydrodynamic equations. We first provide a derivation of this system
from a two-fluids Euler-Maxwell system for electrons and ions, through a set of
scaling limits. We also propose a kinetic formulation for the Hall-MHD
equations which contains as fluid closure different variants of the Hall-MHD
model. Then, we prove the existence of global weak solutions for the
incompressible viscous resistive Hall-MHD model. We use the particular
structure of the Hall term which has zero contribution to the energy identity.
Finally, we discuss particular solutions in the form of axisymmetric purely
swirling magnetic fields and propose some regularization of the Hall equation
Continuum viscoplastic simulation of a granular column collapse on large slopes: Ό(I) rheology and lateral wall effects
We simulate here dry granular flows resulting from the collapse of granular columns on an inclined channel (up to 22°) and compare precisely the results with laboratory experiments. Incompressibility is assumed despite the dilatancy observed in the experiments (up to 10%). The 2-D model is based on the so-called ÎŒ(I) rheology that induces a Drucker-Prager yield stress and a variable viscosity. A nonlinear Coulomb friction term, representing the friction on the lateral walls of the channel, is added to the model. We demonstrate that this term is crucial to accurately reproduce granular collapses on slopes âł10°, whereas it remains of little effect on the horizontal slope. Quantitative comparison between the experimental and numerical changes with time of the thickness profiles and front velocity makes it possible to strongly constrain the rheology. In particular, we show that the use of a variable or a constant viscosity does not change significantly the results provided that these viscosities are of the same order. However, only a fine tuning of the constant viscosity (η=1 Pa s) makes it possible to predict the slow propagation phase observed experimentally at large slopes. Finally, we observed that small-scale instabilities develop when refining the mesh (also called ill-posed behavior, characterized in the work of Barker et al. [âWell-posed and ill-posed behaviour of the ÎŒ(I)-rheology for granular flow,â J. Fluid Mech. 779, 794â818 (2015)] and in the present work) associated with the mechanical model. The velocity field becomes stratified and the bands of high velocity gradient appear. These model instabilities are not avoided by using variable viscosity models such as the ÎŒ(I) rheology. However we show that the velocity range, the static-flowing transition, and the thickness profiles are almost not affected by them
Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids
In this article we study the long-time behaviour of a system of nonlinear
Partial Differential Equations (PDEs) modelling the motion of incompressible,
isothermal and conducting modified bipolar fluids in presence of magnetic
field. We mainly prove the existence of a global attractor denoted by \A for
the nonlinear semigroup associated to the aforementioned systems of nonlinear
PDEs. We also show that this nonlinear semigroup is uniformly differentiable on
\A. This fact enables us to go further and prove that the attractor \A is
of finite-dimensional and we give an explicit bounds for its Hausdorff and
fractal dimensions.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10440-014-9964-
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