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THE Navier-stokes flow around a rotating obstacle with time-dependent body force

Toshiaki Hishida (2009)

Banach Center Publications

We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, under the action of external force f. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain D. Given f = divF with F ∈ B U C ( ℝ ; L 3 / 2 , ∞ ( D ) ) , we consider this problem in D × ℝ and prove that there exists a unique solution u ∈ B U C ( ℝ ; L 3 , ∞ ( D ) ) when F and |ω| are sufficiently small. If, in particular, the external force for...

The nonlinear future stability of the FLRW family of solutions to the irrotational Euler-Einstein system with a positive cosmological constant

Igor Rodnianski, Jared Speck (2013)

Journal of the European Mathematical Society

In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in 1 + 3 spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state p = c 2 ρ , 0 < c 2 < 1 / 3 , the background metric + fluid solutions...

The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes

Jonathan Luk (2013)

Journal of the European Mathematical Society

We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the improved decay rates in the region { r ≤ t 4 } .

The PDE describing constant mean curvature surfaces

Hongyou Wu (2001)

Mathematica Bohemica

We give an expository account of a Weierstrass type representation of the non-zero constant mean curvature surfaces in space and discuss the meaning of the representation from the point of view of partial differential equations.

The polarization in a ferroelectric thin film: local and nonlocal limit problems

Antonio Gaudiello, Kamel Hamdache (2013)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, starting from classical non-convex and nonlocal 3D-variational model of the electric polarization in a ferroelectric material, via an asymptotic process we obtain a rigorous 2D-variational model for a thin film. Depending on the initial boundary conditions, the limit problem can be either nonlocal or local.

The problem of data assimilation for soil water movement

François-Xavier Le Dimet, Victor Petrovich Shutyaev, Jiafeng Wang, Mu Mu (2004)

ESAIM: Control, Optimisation and Calculus of Variations

The soil water movement model governed by the initial-boundary value problem for a quasilinear 1-D parabolic equation with nonlinear coefficients is considered. The generalized statement of the problem is formulated. The solvability of the problem is proved in a certain class of functional spaces. The data assimilation problem for this model is analysed. The numerical results are presented.

The problem of data assimilation for soil water movement

François-Xavier Le Dimet, Victor Petrovich Shutyaev, Jiafeng Wang, Mu Mu (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The soil water movement model governed by the initial-boundary value problem for a quasilinear 1-D parabolic equation with nonlinear coefficients is considered. The generalized statement of the problem is formulated. The solvability of the problem is proved in a certain class of functional spaces. The data assimilation problem for this model is analysed. The numerical results are presented.

The problem of dynamic cavitation in nonlinear elasticity

Jan Giesselmann, Alexey Miroshnikov, Athanasios E. Tzavaras (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

The notion of singular limiting induced from continuum solutions (slic-solutions) is applied to the problem of cavitation in nonlinear elasticity, in order to re-assess an example of non-uniqueness of entropic weak solutions (with polyconvex energy) due to a forming cavity.

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