169 research outputs found
A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations
We consider boundary value problems of the first and third kind for the
diffusionwave equation. By using the method of energy inequalities, we find a
priori estimates for the solutions of these boundary value problems.Comment: 10 pages, no figur
The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow
We propose a mathematical derivation of Brinkman's force for a cloud of
particles immersed in an incompressible fluid. Our starting point is the Stokes
or steady Navier-Stokes equations set in a bounded domain with the disjoint
union of N balls of radius 1/N removed, and with a no-slip boundary condition
for the fluid at the surface of each ball. The large N limit of the fluid
velocity field is governed by the same (Navier-)Stokes equations in the whole
domain, with an additional term (Brinkman's force) that is (minus) the total
drag force exerted by the fluid on the particle system. This can be seen as a
generalization of Allaire's result in [Arch. Rational Mech. Analysis 113
(1991), 209-259] who treated the case of motionless, periodically distributed
balls. Our proof is based on slightly simpler, though similar homogenization
techniques, except that we avoid the periodicity assumption and use instead the
phase-space empirical measure for the particle system. Similar equations are
used for describing the fluid phase in various models for sprays
Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem
We study a system of nonlinear partial differential equations resulting from
the traditional modelling of oil engineering within the framework of the
mechanics of a continuous medium. Recent results on the problem provide
existence, uniqueness and regularity of the optimal solution. Here we obtain
the first necessary optimality conditions.Comment: 9 page
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-
Global Solutions of the Navier-Stokes Equations for Isentropic Flow with Large External Potential Force
We prove the global-in-time existence of weak solutions to the Navier-Stokes
equations of compressible isentropic flow in three space dimensions with
adiabatic exponent . Initial data and solutions are small in
around a non-constant steady state with densities being positive and
essentially bounded. No smallness assumption is imposed on the external forces
when . A great deal of information about partial regularity and
large-time behavior is obtained.Comment: 17 page
Existence of weak solutions for the generalized Navier-Stokes equations with damping
In this work we consider the generalized Navier-Stokes equations with the presence of a damping term in the momentum equation. The problem studied here derives from the set of equations which govern isothermal flows of incompressible and homogeneous non-Newtonian fluids. For the generalized Navier-Stokes problem with damping, we prove the existence of weak solutions by using regularization techniques, the theory of monotone operators and compactness arguments together with the local decomposition of the pressure and the Lipschitz-truncation method. The existence result proved here holds for any and any sigma > 1, where q is the exponent of the diffusion term and sigma is the exponent which characterizes the damping term.MCTES, Portugal [SFRH/BSAB/1058/2010]; FCT, Portugal [PTDC/MAT/110613/2010]info:eu-repo/semantics/publishedVersio
A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem
A wide variety of articles, starting with the famous paper (Gidas, Ni and
Nirenberg in Commun. Math. Phys. 68, 209-243 (1979)) is devoted to the
uniqueness question for the semilinear elliptic boundary value problem
-{\Delta}u={\lambda}u+u^p in {\Omega}, u>0 in {\Omega}, u=0 on the boundary of
{\Omega}, where {\lambda} ranges between 0 and the first Dirichlet Laplacian
eigenvalue. So far, this question was settled in the case of {\Omega} being a
ball and, for more general domains, in the case {\lambda}=0. In (McKenna et al.
in J. Differ. Equ. 247, 2140-2162 (2009)), we proposed a computer-assisted
approach to this uniqueness question, which indeed provided a proof in the case
{\Omega}=(0,1)x(0,1), and p=2. Due to the high numerical complexity, we were
not able in (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)) to treat
higher values of p. Here, by a significant reduction of the complexity, we will
prove uniqueness for the case p=3
On the symmetry of minimizers
For a large class of variational problems we prove that minimizers are
symmetric whenever they are .Comment: 17 pages, to appear in Arch. Rational Mech. Anal.; added Example 7
and some reference
Entire solutions of hydrodynamical equations with exponential dissipation
We consider a modification of the three-dimensional Navier--Stokes equations
and other hydrodynamical evolution equations with space-periodic initial
conditions in which the usual Laplacian of the dissipation operator is replaced
by an operator whose Fourier symbol grows exponentially as \ue ^{|k|/\kd} at
high wavenumbers . Using estimates in suitable classes of analytic
functions, we show that the solutions with initially finite energy become
immediately entire in the space variables and that the Fourier coefficients
decay faster than \ue ^{-C(k/\kd) \ln (|k|/\kd)} for any . The
same result holds for the one-dimensional Burgers equation with exponential
dissipation but can be improved: heuristic arguments and very precise
simulations, analyzed by the method of asymptotic extrapolation of van der
Hoeven, indicate that the leading-order asymptotics is precisely of the above
form with . The same behavior with a universal constant
is conjectured for the Navier--Stokes equations with exponential
dissipation in any space dimension. This universality prevents the strong
growth of intermittency in the far dissipation range which is obtained for
ordinary Navier--Stokes turbulence. Possible applications to improved spectral
simulations are briefly discussed.Comment: 29 pages, 3 figures, Comm. Math. Phys., in pres
Global Solutions for Incompressible Viscoelastic Fluids
We prove the existence of both local and global smooth solutions to the
Cauchy problem in the whole space and the periodic problem in the n-dimensional
torus for the incompressible viscoelastic system of Oldroyd-B type in the case
of near equilibrium initial data. The results hold in both two and three
dimensional spaces. The results and methods presented in this paper are also
valid for a wide range of elastic complex fluids, such as magnetohydrodynamics,
liquid crystals and mixture problems.Comment: We prove the existence of global smooth solutions to the Cauchy
problem for the incompressible viscoelastic system of Oldroyd-B type in the
case of near equilibrium initial dat
- …