Persistence of undercompressive phase boundaries for isothermal Euler equations including configurational forces and surface tension

The persistence of subsonic phase boundaries in a multidimensional Van der Waals fluid is analyzed. The phase boundary is considered as a sharp free boundary that connects liquid and vapor bulk phase dynamics given by the isothermal Euler equations. The evolution of the boundary is driven by effects of configurational forces as well as surface tension. To analyze this problem, the equations and trace conditions are linearized such that one obtains a general hyperbolic initial boundary value problem with higher‐order boundary conditions. A global existence theorem for the linearized system with constant coefficients is shown. The proof relies on the normal mode analysis and a linear form in suitable spaces that is defined using an associated adjoint problem. Especially, the associated adjoint problem satisfies the uniform backward in time Kreiss–Lopatinski? condition. A new energy‐like estimate that also includes surface energy terms leads finally to the uniqueness and regularity for the found solutions of the problem in weighted spaces. Copyright © 2016 John Wiley & Sons, Ltd.     

Symmetrizer and Antisymmetrizer of the Birman–Wenzl–Murakami Algebras

Explicit formulas for the symmetrizer and the antisymmetrizer of the Birman–Wenzl–Murakami algebras BWM(r,q) _n are given.     

Computing matrix symmetrizers,finally possible via the Huang and Nong algorithm

By a theorem of Frobenius (F.G. Frobenius, Über die mit einer Matrix vertauschbaren Matrizen, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin (1910), pp. 3–15 (also in Gesammelte Abhandlungen, Band 3, Springer 1968. pp. 415–427)), every matrix A _n,n over any field 𝔽 is the product of two symmetric ones. Using the algorithm of Huang and Nong (J. Huang and L. Nong, An iterative algorithm for solving finite-dimensional linear operator equations T(x)?=?f with applications, Linear Algebra Appl. 432 (2010), pp. 1176–1188) for linear systems, we develop an algorithm to compute a symmetric matrix S?=?S ^T ?∈?𝔽 ^n,n for which SA is symmetric for any given square matrix A?∈?𝔽 ^n,n where 𝔽?=?? or ?. The algorithm is implemented and tested in MATLAB.     

Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrixX that satisfies the equationXA =A ^tX, wheret indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.     

First Fundamental Theorem for Covariants of Classical Groups

Let U(G) be a maximal unipotent subgroup of one of the classical groups G=GL(V), O(V), Sp(V). Let W be a direct sum of copies of V and its dual V*. For the natural action U(G) : W, we describe a minimal system of homogeneous generators for the algebra of U(G)-invariant regular functions on W. For G=O(V), Sp(V), this result is connected with a construction for the irreducible representations of G due to H. Weyl.     

Stability of an incompressible plasma–vacuum interface with displacement current in vacuum

We study the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement when the vacuum magnetic field obeys the div-curl system of pre-Maxwell dynamics, to better understand the influence of the electric field in vacuum, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Under the necessary and sufficient stability condition for a planar interface found earlier by Trakhinin, we prove an energy a priori estimate for the linearized constant coefficient problem. The process of derivation of this estimate is based on various methods, including a secondary symmetrization of the vacuum Maxwell equations, the derivation of a hyperbolic evolutionary equation for the interface function, and the construction of a degenerate Kreiss-type symmetrizer for an elliptic-hyperbolic problem for the total pressure.     

A Kreiss' Symmetrizer Domain Penalization Method

In this paper, we describe a new way to approximate solutions of mixed hyperbolic systems with constant coefficients satisfying a Uniform Lopatinski Condition via a domain penalization method. The construction of the penalization operator is based on a Kreiss' symmetrizer. The obtained method has the advantage of not generating any boundary layers.