408 research outputs found
An optimal gap theorem
By solving the Cauchy problem for the Hodge-Laplace heat equation for
-closed, positive -forms, we prove an optimal gap theorem for
K\"ahler manifolds with nonnegative bisectional curvature which asserts that
the manifold is flat if the average of the scalar curvature over balls of
radius centered at any fixed point is a function of .
Furthermore via a relative monotonicity estimate we obtain a stronger
statement, namely a `positive mass' type result, asserting that if is
not flat, then for any
Lattice gas model in random medium and open boundaries: hydrodynamic and relaxation to the steady state
We consider a lattice gas interacting by the exclusion rule in the presence
of a random field given by i.i.d. bounded random variables in a bounded domain
in contact with particles reservoir at different densities. We show, in
dimensions , that the rescaled empirical density field almost surely,
with respect to the random field, converges to the unique weak solution of a
non linear parabolic equation having the diffusion matrix determined by the
statistical properties of the external random field and boundary conditions
determined by the density of the reservoir. Further we show that the rescaled
empirical density field, in the stationary regime, almost surely with respect
to the random field, converges to the solution of the associated stationary
transport equation
Global Stability of a Premixed Reaction Zone (Time-Dependent Liñan’s Problem)
Global stability properties of a premixed, three-dimensional reaction zone are considered. In the nonadiabatic case (i.e., when there is a heat exchange between the reaction zone and the burned gases) there is a unique, spatially one-dimensional steady state that is shown to be unstable (respectively, asymptotically stable) if the reaction zone is cooled (respectively, heated) by the burned mixture. In the adiabatic case, there is a unique (up to spatial translations) steady state that is shown to be stable. In addition, the large-time asymptotic behavior of the solution is analyzed to obtain sufficient conditions on the initial data for stabilization. Previous partial numerical results on linear stability of one-dimensional reaction zones are thereby confirmed and extended
Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations
We present new interior regularity criteria for suitable weak solutions of
the 3-D Navier-Stokes equations: a suitable weak solution is regular near an
interior point if either the scaled -norm of the velocity
with , , or the -norm of the
vorticity with , , or the
-norm of the gradient of the vorticity with , , , is sufficiently small near
Simultaneous determination of time-dependent coefficients and heat source
This article presents a numerical solution to the inverse problems of simultaneous determination of the time-dependent coefficients and the source term in the parabolic heat equation subject to overspecified conditions of integral type. The ill-posed problems are numerically discretized using the finite-difference method. The resulting system of nonlinear equations is solved numerically using the MATLAB toolbox routine lsqnonlin applied to minimizing the nonlinear Tikhonov regularization functional subject to simple physical bounds on the variables. Numerical examples are presented to illustrate the accuracy and stability of the solution
Well-posedness for a model of individual clustering
25 pagesInternational audienceWe study the well-posedness of a model of individual clustering. Given p > N ≥ 1 and an initial condition in W 1,p (Ω), the local existence and uniqueness of a strong solution is proved. We next consider two specific reproduction rates and show global existence if N = 1, as well as, the convergence to steady states for one of these rates
An alternative approach to regularity for the Navier-Stokes equations in critical spaces
In this paper we present an alternative viewpoint on recent studies of
regularity of solutions to the Navier-Stokes equations in critical spaces. In
particular, we prove that mild solutions which remain bounded in the space
do not become singular in finite time, a result which was proved
in a more general setting by L. Escauriaza, G. Seregin and V. Sverak using a
different approach. We use the method of "concentration-compactness" +
"rigidity theorem" which was recently developed by C. Kenig and F. Merle to
treat critical dispersive equations. To the authors' knowledge, this is the
first instance in which this method has been applied to a parabolic equation.
We remark that we have restricted our attention to a special case due only to
a technical restriction, and plan to return to the general case (the
setting) in a future publication.Comment: 41 page
On the well-posedness of mathematical models for multicomponent biofilms
Bacterial biofilms are microbial depositions on immersed surfaces. Their mathematical description leads to degenerate diffusion-reaction equations with two non-Fickian effects: (i) a porous medium equation like degeneracy where the biomass density vanishes and (ii) a super-diffusion singularity if the biomass density reaches its threshold density. In the case of multispecies interactions, several such equations are coupled, both in the reaction terms and in the nonlinear diffusion operator. In this paper, we generalize previous work on existence and uniqueness of solutions of this type of models and give a general, relatively easy to apply criterion for well-posedness. The use of the criterion is illustrated in several examples from the biofilm modeling literature
Simultaneous determination of time and space-dependent coefficients in a parabolic equation
This paper investigates a couple of inverse problems of simultaneously determining time and space dependent coefficients in the parabolic heat equation using initial and boundary conditions of the direct problem and overdetermination conditions. The measurement data represented by these overdetermination conditions ensure that these inverse problems have unique solutions. However, the problems are still ill-posed since small errors in the input data cause large errors in the output solution. To overcome this instability we employ the Tikhonov regularization method. The finite-difference method (FDM) is employed as a direct solver which is fed iteratively in a nonlinear minimization routine. Both exact and noisy data are inverted. Numerical results for a few benchmark test examples are presented, discussed and assessed with respect to the FDM mesh size discretisation, the level of noise with which the input data is contaminated, and the chosen regularization parameters
Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion
Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters
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