Stability and non-normality of the k-ε equations

The analytical and numerical solutions of the equations of the k-ε turbulence model are analyzed. Under certain conditions on the boundary values and the interior values of k and ε the analytical and numerical solutions are bounded. If the steady state solution is obtained numerically by a Runge-Kutta time-stepping method, then severe constraints on the time-step and the non-normality of the jacobian matrix make the convergence very slow. The simplifications and conclusions are supported by data from a numerical solution of flow over a flat plate.     

On the relation between the continuous and discrete Painlevé equations

A method for deriving difference equations (the discrete Painlevé equations in particular) from the Bäcklund transformations of the continuous Painlevé equations is discussed. This technique can be used to derive several of the known discrete painlevé equations (in particular, the first and second discrete Painlevé equations and some of their alternative versions). The Painlevé equations possess hierarchies of rational solutions and one-parameter families of solutions expressible in terms of the classical special functions for special values of the parameters. Hence, the aforementioned relations can be used to generate hierarchies of exact solutions for the associated discrete Painlevé equations. Exact solutions of the Painlevé equations simultaneously satisfy both a differential equation and a difference equation, analogously to the special functions.     

Well-posedness for the system of the Hamilton–Jacobi and the continuity equations

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S _t + H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of

Convergence and continuous dependence for the Brinkman–Forchheimer equations

This paper investigates the flow of fluid in a porous medium which is described in the Brinkman–Forchheimer equations and obtains the structural stability results for the coefficients.     

Hamiltonians associated with the third and fifth Painlevé equations

We obtain a Painlevé-type differential equation for the simplest rational Hamiltonian associated with the fifth Painlevé equation in the case γ ≠ 0, δ = 0. We prove the existence of Hamiltonians of a nonrational type associated with the fifth Painlevé equation in the case γ ≠ 0, δ = 0. We obtain a generalization of the Garnier and Okamoto formulas for rational Hamiltonians associated with the third Painlevé tequation.     

Block and joint ℋ︁-matrix preconditioners for the Oseen equations

For saddle point problems in fluid dynamics, many preconditioners in the literature exploit the block structure of the problem to construct block diagonal or block triangular preconditioners. The performance of such preconditioners depends on whether fast, approximate solvers for the linear systems on the block diagonal as well as for the Schur complement are available. We will construct these efficient preconditioners using hierarchical matrix techniques in which fully populated matrices are approximated by blockwise low rank approximations. We will compare such block preconditioners with those obtained through a completely different approach where the given block structure is not used but a domain-decomposition based ℋ︁-LU factorization is constructed for the complete system matrix. Preconditioners resulting from these two approaches will be discussed and compared through numerical results. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Poincaré-Steklov integral equations and the Riemann monodromy problem

We consider the Poincaré-Steklov singular integral equation obtained by reducing a boundary value problem for the Laplace operator with a spectral parameter in the boundary condition to the boundary. It is shown that this equation can be restated equivalently in terms of the classical Riemann monodromy problem. Several equations of this type are solved in elliptic functions. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow Institute of Physical Engineering (Physical-Technical). Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 9–22, April–June, 2000. Translated by V. E. Nazaikinskii     

Liouville theorems for the Navier–Stokes equations and applications

We study bounded ancient solutions of the Navier–Stokes equations. These are solutions with bounded velocity defined in R ⁿ × (−1, 0). In two space dimensions we prove that such solutions are either constant or of the form u(x, t) = b(t), depending on the exact definition of admissible solutions. The general 3-dimensional problem seems to be out of reach of existing techniques, but partial results can be obtained in the case of axisymmetric solutions. We apply these results to some scenarios of potential singularity formation for axi-symmetric solutions, and obtain extensions of results in a recent paper by Chen, Strain, Tsai and Yau [4].     

Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold

A classification of solutions of the first and second Painlevé equations corresponding to a special distribution of poles at infinity is considered. The relation between this distribution and singularities of the two-dimensional complex monodromy data manifold used for the parameterization of the solutions is analyzed. It turns out that solutions of the Painlevé equations have no poles in a certain critical sector of the complex plane if and only if their monodromy data lie in the singularity submanifold. Such solutions belong to the so-called class of “truncated” solutions (intégrales tronquée) according to P. Boutroux’s classification. It is shown that all known special solutions of the first and second Painlevé equations belong to this class.     

On the theory of Besov–Herz spaces and Euler equations

Inspired by Kalton and Wood’s work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier–Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism T between the associated Fourier algebras (resp. Fourier–Stieltjes algebras) with completely bounded norm \({\left\| T \right\|_{cb}} < \sqrt {3/2} \left( {{\text{resp}}{\text{.}}{{\left\| T \right\|}_{cb}} < \sqrt {5/2} } \right)\). We show similar results involving the norm distortion ‖T‖‖T ^?1‖ with universal but non-explicit bound. Our results subsume Walter’s well-known structural theorems and also Lau’s theorem on the second conjugate of Fourier algebras.     

Some generalization of Cauchy’s and the quadratic functional equations

We find the solutions ${f,g,h \colon S \to H}$ of each of the functional equations $$\sum\limits_{\lambda \in \Lambda} f(x+\lambda y)=|\Lambda| g(x)+h(y),\quad x,y\in S,$$ where (S,?+) is an abelian semigroup, Λ is a finite subgroup of the automorphism group of S,?(H,?+) is an abelian group.     

Transport solutions of the Lamé equations and shock elastic waves

The Lamé system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, or supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Numerical results concerning the dynamics of an elastic medium influenced by concentrated transport loads moving at sub-, tran- and supersonic speeds are presented.