ON SINGULAR SOLUTIONS OF THE STATIONARY NAVIER-STOKES SYSTEM IN POWER CUSP DOMAINS

The boundary value problem for the steady Navier–Stokes system is considered in a 2D bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with a nonzero flow rate is studied. In this case there is a source/sink in O and the solution necessarily has an infinite Dirichlet integral. The formal asymptotic expansion of the solution near the singular point is constructed and the existence of a solution having this asymptotic decomposition is proved.

Formal oscillatory distributions

The formal asymptotic expansion of an oscillatory integral whose phase function has one nondegenerate critical point is a formal distribution supported at the critical point which is applied to the amplitude. This formal distribution is called a formal oscillatory integral (FOI). We introduce the notion of a formal oscillatory distribution supported at a point. We prove that a formal distribution is given by some FOI if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We also prove that a star product ⋆ on a Poisson manifold M is natural in the sense of Gutt and Rawnsley if and only if the formal distribution f ⊗ g ↦ ( f ⋆ g ) ( x ) is oscillatory for every x ∈ M.

Homogenizations of integro-differential equations with Lévy operators with asymmetric and degenerate densities

We consider periodic homogenization problems for Lévy operators with asymmetric Lévy densities. The formal asymptotic expansion used for the α-stable (symmetric) Lévy operators (α ∈ (0, 2)) is not directly applicable to such asymmetric cases. We rescale the asymmetric densities and extract the most singular parts of the measures, which average out the microscopic dependencies in the homogenization procedures. We give two conditions, (A) and (B), that characterize such a class of asymmetric densities under which the above ‘rescaled’ homogenization is available.

Asymptotic expansions for distributions of sums of a double sequence of quasilattice random variables

We consider the formal asymptotic expansion of probability distribution of the sums of independent random variables. The approximation was made by using infinitely divisible probability distributions.

CONVERGENCE FOR LONG-TIMES OF A SEMIDISCRETE PERONA–MALIK EQUATION IN ONE DIMENSION

We rigorously prove that the semidiscrete schemes of a Perona–Malik type equation converge, in a long-time scale, to a suitable system of ordinary differential equations defined on piecewise constant functions. The proof is based on a formal asymptotic expansion argument, and on a careful construction of discrete comparison functions. Despite the equation having a region where it is backward parabolic, we prove a discrete comparison principle, which is the key tool for the convergence result.

ASYMPTOTIC ESTIMATES FOR THE MULTI-DIMENSIONAL ELECTRO-DIFFUSION EQUATIONS

The drift-diffusion equations can be studied in the framework of singular perturbation analysis as a small parameter characterizing the device goes to zero. A formal asymptotic expansion, which includes internal (and eventually boundary) layer terms can be derived by standard techniques of asymptotic analysis. We present in this paper L2 estimates for the difference between the solutions of the full system and the first term of the expansion which are valid for multi-dimensional devices close to equilibrium. These estimates are based on a uniform monotone property of the equations with respect to the small parameter and on a L∞ bound for the gradient of solutions of equations under divergence form whose coefficients are bounded, and have derivatives in one direction which are bounded with respect to all variables.

On two-dimensional slow viscous flows past obstacles in a half-plane

SynopsisWe consider a cylinder with arbitrary cross section moving in a viscous incompressible fluid parallel to a plane wall. Formal asymptotic expansions of the solution for small Reynolds numbers are constructed by using boundary integral equations of the first kind. In contrast to the problem without a wall, we show that there exists a unique solution to the zeroth order problem. However, the problem considered here is still singular in the sense that we find the Stokes paradox in the next higher order problem. A justification of the formal asymptotic expansion for the first two terms is established rigorously.

Oscillating Progressive Shear Waves in Nonhomogeneous Viscoelastic Solids

The equations of motion for cylindrical and spherical shear waves in nonhomogeneous, isotropic, "standard", viscoelastic media with continuous radial variations are derived. Oscillatory shearing tractions are applied to the boundaries of cylindrical and spherical openings in unlimited viscoelastic media. The propagation of small-amplitude waves is studied, and formal asymptotic expansions of the solutions are obtained. In both cases (cylindrical and spherical), the leading term of the formal asymptotic expansion represents a modulated, oscillating, progressive wave propagating with variable velocity. The modulation depends on the moduli of rigidity and viscosity, whereas the velocity depends on the moduli of rigidity only. Application of our results to the propagation of shear waves in both finite and infinite viscoelastic plates is discussed.