Toroidal Spaces Research Articles

The two‐dimensional stationary Navier–Stokes equations in toroidal Besov spaces

AbstractWe consider the stationary Navier–Stokes equations in the two‐dimensional torus . For any , we show the existence, uniqueness, and continuous dependence of solutions in homogeneous toroidal Besov spaces for given small external forces in when . These spaces become closer to the scaling invariant ones if the difference ε becomes smaller. This well‐posedness is proved by using the embedding property and the para‐product estimate in homogeneous Besov spaces. In addition, for the case , we can show the ill‐posedness, even in the scaling invariant spaces. Actually in such cases of p and q, we can prove that ill‐posedness by showing the discontinuity of a certain solution map from to .

Duality of moduli in regular toroidal metric spaces

We generalize a result of Freedman and He [4, Theorem 2.5], concerning the duality of moduli and capacities in solid tori, to sufficiently regular metric spaces. This is a continuation of the work of the author and Rajala [12] on the corresponding duality in condensers.

Well-posedness and ill-posedness of the stationary Navier–Stokes equations in toroidal Besov spaces

We consider the stationary Navier–Stokes equations in the n-dimensional torus for . We show the existence and uniqueness of solutions in homogeneous toroidal Besov spaces with for small external forces in . We can show its well-posedness by a similar method to that of Kaneko–Kozono–Shimizu (Indiana Univ. Math. J.), which has investigated the same problem in homogeneous Besov spaces on . Our advantage is to prove the ill-posedness in the critical exponents like p = n, and , . Indeed in such cases of p and q, there exists a sequence of external forces which converges to zero in and yields a sequence of solutions which does not converge to zero in . We can show this ill-posedness by constructing the sequence of external forces, as similar to those of initial data proposed by Yoneda (2010 J. Funct. Anal. 258 3376–87) in the non-stationary problem.

Some new results for the one-loop mass correction to the compactified λϕ4 theory

In this work, we consider the one-loop effective action of a self-interacting λϕ4 field propagating in a D dimensional Euclidean space endowed with d ≤ D compact dimensions. The main purpose of this paper is to compute the corrections to the mass of the field due to the presence of the compactified dimensions. Although the results of the one-loop correction to the mass of a λϕ4 field are very well known for compactified toroidal spaces, where the field obeys periodic boundary conditions, similar results do not appear to be readily available for cases in which the scalar field is subject to Dirichlet and Neumann boundary conditions. We apply the results of the one-loop mass correction to the study of the critical temperature in Ginzburg-Landau models.

Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaces

In this paper, we consider pseudodifferential operators on the torus with operator-valued symbols and prove continuity properties on vector-valued toroidal Besov spaces, without assumptions on the underlying Banach spaces. The symbols are of limited smoothness with respect to $x$ and satisfy a finite number of estimates on the discrete derivatives. The proof of the main result is based on a description of the operator as a convolution operator with a kernel representation which is related to the dyadic decomposition appearing in the definition of the Besov space.

Finite-size, chemical-potential and magnetic effects on the phase transition in a four-fermion interacting model

We study effects coming from finite size, chemical potential and from a magnetic background on a massive version of a four-fermion interacting model. This is performed in four dimensions as an application of recent developments for dealing with field theories defined on toroidal spaces. We study effects of the magnetic field and chemical potential on the size-dependent phase structure of the model, in particular, how the applied magnetic field affects the size-dependent critical temperature. A connection with some aspects of the hadronic phase transition is established.

CASIMIR FORCE OF FERMIONS COUPLED TO MONOPOLES IN SIX-DIMENSIONAL SPACETIME

We calculate the Casimir force for a fermionic quantum field in a piston geometry with three parallel plates. The fermion satisfies bag boundary conditions on the plates and the spacetime is assumed to have compact extra dimensions. The calculation is performed in the cases where the extra space has toroidal and spherical topology. We are mainly interested in the case in which the fermion is coupled non-trivially to an extra-dimensional defect, with a torus extra-dimensional topological background. We found that in certain limits, the Casimir force corresponding to the defect-fermion system and to the sphere, has opposite sign, in reference to those corresponding to the toroidal extra-dimensional spaces.

Finite-size, magnetic and chemical-potential effects on first-order phase transitions

Abstract We perform a study about effects of an applied magnetic field and a finite chemical potential on the size-dependent phase structure of a first-order transition. These effects are introduced by using methods of quantum fields defined on toroidal spaces, and we study in particular the case of two compactified dimensions, imaginary time and a spatial one (a heated film). It is found that for any value of the applied field, there is a minimal size of the system, independent of the chemical potential, below which the transition disappears.