Quasi-static Maxwell's equations with a dissipative non-linear boundary condition: Full discretization

We study a time dependent eddy current equation for the magnetic field H accompanied with a nonlinear boundary condition, which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electric (E) and magnetic (H  ) fields obeys the following power law (linearized for small and large values) ν×E=ν×(|H×ν|^α−1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times ({|\mathit{H}\times \mathit{\nu }|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We design a linear fully discrete approximation scheme to solve this nonlinear problem. The convergence of the approximations to a weak solution is proved, error estimates describing the dependence of the error on discretization parameters are derived as well. The efficiency of the proposed method is supported by numerical experiments.     

Time-discretization scheme for quasi-static Maxwell's equations with a non-linear boundary condition

We study a time dependent eddy current equation for the magnetic field H$\mathit{H}$ accompanied with a non-linear degenerate boundary condition (BC), which is a generalization of the classical Silver–Müller condition for a non-perfect conductor. More exactly, the relation between the normal components of electrical E$\mathit{E}$ and magnetic H$\mathit{H}$ fields obeys the following power law ν×E=ν×(|H×ν|^α-1H×ν)$\mathit{\nu }\times \mathit{E}=\mathit{\nu }\times (|\mathit{H}\times \mathit{\nu }{|}^{\alpha -1}\mathit{H}\times \mathit{\nu })$ for some α∈(0,1]$\alpha \in (0,1]$. We establish the existence and uniqueness of a weak solution in a suitable function space under the minimal regularity assumptions on the boundary Γ$\Gamma$ and the initial data _H0${\mathit{H}}_0$. We design a non-linear time discrete approximation scheme based on Rothe's method and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time-discretization.     

The boundary element solution of the magnetohydrodynamic flow in an infinite region

We consider the magnetohydrodynamic (MHD) flow which is laminar, steady and incompressible, of a viscous and electrically conducting fluid on the half plane (y≥0)$(y\ge 0)$. The boundary y=0$y=0$ is partly insulated and partly perfectly conducting. An external circuit is connected so that current enters the fluid at discontinuity points through external circuits and moves up on the plane. The flow is driven by the interaction of imposed electric currents and a uniform, transverse magnetic field applied perpendicular to the wall, y=0$y=0$. The MHD equations are coupled in terms of the velocity and the induced magnetic field. The boundary element method (BEM) is applied here by using a fundamental solution which enables treating the MHD equations in coupled form with general wall conditions. Constant elements are used for the discretization of the boundary y=0$y=0$ only since the boundary integral equation is restricted to this boundary due to the regularity conditions at infinity. The solution is presented for the values of the Hartmann number up to M=700$M=700$ in terms of equivelocity and induced magnetic field contours which show the well-known characteristics of the MHD flow. Also, the thickness of the parabolic boundary layer propagating in the field from the discontinuity points in the boundary conditions, is calculated.     

Linear spaces of matrices of constant rank and instanton bundles

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P³${\mathbb{P}}^3$ and derived category tools. The method allows one to prove the existence of new examples of size 10×10$10\times 10$ and 14×14$14\times 14$ via instanton bundles of charge 2 and 4 respectively, and it provides an explanation for what used to be the only known example (Westwick 1996 [25]). We also give an algorithm to construct explicitly a matrix of size 14 of this type.     

On singular foliations on the solid torus

We study smooth foliations on the solid torus S¹×D²$S^1\times D^2$ having S¹×{0}$S^1\times \{\mathbf{0}\}$ and S¹×∂D²$S^1\times \partial D^2$ as the only compact leaves and S¹×{0}$S^1\times \{\mathbf{0}\}$ as singular set. We show that all other leaves can only be cylinders or planes, and give necessary conditions for the foliation to be a suspension of a diffeomorphism of the disc.     

Triangularizing matrix polynomials

For an algebraically closed field F$\mathbb{F}$, we show that any matrix polynomial P(λ)∈F[λ]^n×m$P(\lambda )\in \mathbb{F}{[\lambda ]}^{n\times m}$, n?m$n?m$, can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors. We also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes 1×1$1\times 1$ and 2×2$2\times 2$. The proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors.     

Subharmonic solutions bifurcated from homoclinic orbits for weakly coupled singular systems

The problem of bifurcation from homoclinic solution towards periodic solution was considered for weekly coupled singular systems. By using functional analytic approach based on the Lyapunov–Schmidt reduction, we obtained some functions H:R^d-1×R→R^d$H:R^{d-1}\times R\to R^d$. The simple roots of the equations, H(α,β)=0$H(\alpha ,\beta )=0$, correspond to the existence of subharmonic solutions. And if the vector field is 2-period, then for any integer m  , the weakly coupled singular system has 2m$2m$-period solution.     

An operator-theoretical treatment of the Maskawa–Nakajima equation in the massless abelian gluon model

The Maskawa–Nakajima equation has attracted considerable interest in elementary particle physics. From the viewpoint of operator theory, we study the Maskawa–Nakajima equation in the massless abelian gluon model. We first show that there is a nonzero solution to the Maskawa–Nakajima equation when the parameter λ   satisfies λ>2$\lambda >2$. Moreover, we show that the solution is infinitely differentiable and strictly decreasing. We thus conclude that the massless abelian gluon model generates the nonzero quark mass spontaneously and exhibits the spontaneous chiral symmetry breaking when λ>2$\lambda >2$. We next show that there is a unique solution 0 to the Maskawa–Nakajima equation when 0<λ<1$0<\lambda <1$, from which we conclude that each quark remains massless and that the model realizes the chiral symmetry when 0<λ<1$0<\lambda <1$.     

Time dependent random fields on spherical non-homogeneous surfaces

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere which is represented by a time-changed spherical Brownian motion of order ν∈(0,1]$\nu \in (0,1]$. We can capture some anisotropies in Cosmology with this model. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order ν$\nu$. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotropic random field is indexed. The time dependent random fields we present in this work are therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.     

Strong unique continuation for systems of complex vector fields

In this work we study the property of strong unique continuation, at a given point, for Gevrey solutions to homogeneous systems of PDE defined by complex, real-analytic vector fields in involution. We show that when the system is minimal at the point then the strong unique continuation property holds for Gevrey solutions of order σ∈[1,2]$\sigma \in [1,2]$ and, furthermore, when the minimality property fails to hold then there are non-trivial Gevrey flat solutions of any given order σ>1$\sigma >1$. The case of Gevrey order σ>2$\sigma >2$ is also studied for some particular classes of involutive systems.     

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R²)$\theta \in C^{\delta }({\mathbb{R}}^2)$ with δ>1−2α$\delta >1-2\alpha$ on the time interval [t₀,t]$[t_0,t]$, then it is actually a classical solution on (t₀,t]$(t_0,t]$.     

Nonexistence of classical steady Hele–Shaw bubble

This paper concerns nonexistence of steadily translating bubbles in a Hele–Shaw cell for small but nonzero surface tension. It is proved that for any fixed bubble velocity U$U$ relative to the fluid velocity at infinity in the interval (2,∞)$(2,\infty )$, there can be no classical steady bubble solution.     

On minors of the compound matrix of a Laplacian

Let L   be an n×n$n\times n$ matrix with zero row and column sums, n?3$n?3$. We obtain a formula for any minor of the (n−2)$(n-2)$-th compound of L. An application to counting spanning trees extending a given forest is given.     

On control of Sobolev norms for some semilinear wave equations with localized data

Consider the semilinear wave equations in dimension 3 with a defocusing and superconformal power-type nonlinearity and with data lying in the H^s×H^s−1$H^s\times H^{s-1}$ (s<1$s<1$) closure of smooth functions that are compactly supported inside a ball with fixed radius. We establish new bounds of the Sobolev norms of the solution. In particular, we prove that the H^s$H^s$ norm of the high frequency component of the solution grows like T^∼(1−s)2+$T^{\sim {(1-s)}^2+}$ in a neighborhood of s=1$s=1$. In order to do that, we perform an analysis in a neighborhood of the cone, using the finite speed of propagation, an almost Shatah–Struwe estimate [17], an almost conservation law and a low–high frequency decomposition  and .¹     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.     

Singular solutions of Hessian elliptic equations in five dimensions

We show that for any δ∈[0,1)$\delta \in [0,1)$ there exists a homogeneous order 2−δ$2-\delta$ analytic outside zero solution to a uniformly elliptic Hessian equation in R⁵${\mathbb{R}}^5$.     

Regularity of the solutions to the liquid crystal equations with small rough data

In [30], the author has proved the existence of Koch–Tataru [12] type solution to the incompressible nematic liquid crystal flow with initial data (_u0,_d0)$(u_0,d_0)$ in BMO⁻¹×BMO${\mathit{BMO}}^{-1}\times \mathit{BMO}$. In this paper, we shall present that the solution in [30] has arbitrary spatial regularity. Furthermore, we shall give the decay estimates for any space derivative.