Singular Multipliers Research Articles

Large time behaviors of strong solutions to magnetohydrodynamic equations with free boundary and degenerate viscosity

In this paper, we establish the local, global existence and large-time behaviors of strong solutions to the free boundary problem of the planar magnetohydrodynamic equations with degenerate viscosity coefficient. Only the initial energy at the basic level is required to be small. The main difficulties are the degeneracy of the system near the free boundary and the strong coupling of the magnetic field and the velocity. We overcome the trouble by deriving the point-wise upper and lower bounds of the deformation variable uniformly in time and spatial variables and setting up the uniform-in-time weights energy estimates of solutions via singular multipliers. In contrast to previous works, the density is not required to be bounded from below and the viscosity coefficient is not a constant but degenerate; moreover, sharp convergent rates toward the steady state of the solutions are obtained.

A comparison of conservation law construction approaches for the two-dimensional incompressible Mooney-Rivlin hyperelasticity model

An extended Kovalevskaya form is derived for the two-dimensional incompressible Mooney-Rivlin nonlinear hyperelasticity equations and is used to compute a complete set of local conservation laws of the model through the direct method. Conserved densities and fluxes of the conservation laws are derived, and their physical interpretation is discussed. Since the model admits a variational formulation, the equations are rewritten in the self-adjoint form. Computation of local conservation laws through the direct method applied to the self-adjoint form, as well as a conservation law computation through the local symmetry analysis and the Noether’s first theorem, is performed. A correspondence between local variational symmetries and conservation law multipliers is illustrated. It is argued that even though it leads to more complicated forms of multipliers, the direct conservation law construction method applied to the Kovalevskaya form of the equations is a preferred systematic way of conservation law computations for complicated physical models of the type considered in this work, since it yields complete results, and naturally avoids singular multipliers.

Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol e i | ξ | α , where α ∈ [ 0 , 2 ] , are bounded on all modulation spaces, but, in general, fail to be bounded on the usual L p -spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers | ξ | − δ sin ( | ξ | α ) for 0 ⩽ δ ⩽ α .

Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators

Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so-called Coifman-Meyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multipliers generalizations as well as to operators with variable dependent symbols. A symbolic calculus for the transposes of bilinear pseudodifferential operators and for the composition of linear and bilinear pseudodifferential operators is presented too.

A nonconvex separation property and some applications

In this paper we proved a nonconvex separation property for general sets which coincides with the Hahn-Banach separation theorem when sets are convexes. Properties derived from the main result are used to compute the subgradient set to the distance function in special cases and they are also applied to extending the Second Welfare Theorem in economics and proving the existence of singular multipliers in Optimization.

Multi-linear operators given by singular multipliers

We prove L p L^p estimates for a large class of multi-linear operators, which includes the multi-linear paraproducts studied by Coifman and Meyer (1991), as well as the bilinear Hilbert transform and other operators with large groups of modulation symmetries.

A nonconvex separation property in Banach spaces

We establish, in infinite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers and show the relation of our property with the extremal principle of Mordukhovich.

Pointwise maximum principle for convex optimal control problems with mixed control-phase variable inequality constraints

The validity of a global pointwise maximum principle is proved for a class of convex optimal control problems with mixed control-phase variable inequality constraints. No compatibility hypotheses are required, and singular multipliers are avoided.

Stochastic convex programming: singular multipliers and extended duality singular multipliers and duality

A two-stage stochastic programming problem with recourse is studied here in terms of an extended Lagrangian function which allows certain multipliers to be elements of a dual space (i?00)*, rather than an ϊ£λ space. Such multipliers can be decomposed into an i^-component and a component. The generalization makes it possible to characterize solutions to the problem in terms of a saddle-point, if the problem is strictly feasible. The Kuhn-Tucker conditions for the basic duality framework are modified to admit singular multipliers. It is shown that the optimal multiplier vectors in the extended dual problem are, in at least one broad case, ideal limits of maximizing sequences of multiplier vectors in the basic dual problem.