Deformation of free boundaries in plane viscous creep

Summary and Final Comments The deformation of a cylindrical free boundary lying in a thick infinite viscous body loaded by uniform stresses at infinity has been investigated. It is shown that when the free boundary can be initially conformally mapped onto a circle by a polynomial of degreen, it deforms so that its configuration remains within the family of curves described by a polynomial mapping of exactly degreen. A method of obtaining the exact motion of boundaries in this class is formulated and an example illustrating the calculations is offered; the example shows that a small irregularity on a free boundary may grow when compression is applied at infinity.The fact that polynomial free boundaries remain polynomial curves during deformation is one aspect of the kinematic reversibility (or memory) of linear viscous deformation, which greatly facilitates analysis of their motion. The deformation of completely arbitrary free boundaries is more difficult to obtain. In plane elasticity one often seeks approximate solutions to a problem by using a polynomia approximation to the exact mapping of a boundary.Muskhelishvili [3] has shown that if one chooses a sequence {_n} of mappings which converges onto the true mapping of the boundary , the sequences of stress functions {_n} and {_n} generated by {_n} converge onto the stress functions and associated with . In the viscous case Muskhelishvili's theorem always applies for computation of incipient motion; but, (21) and (26) show that when finite motion is considered it is not always adequate to replace the true mapping of the surface with a polynomial which is initially a close approximation. The general conditions under which one may expect the magnitude of higher order terms in the mapping of the free surface to diminish during deformation, so that a polynomial approximation to a mapping improves with time, have not been worked out in detail; however, it seems sufficient at this writing to require that the hydrostatic component of stress at infinity be tensile (P ₁+P₂>0). In the case that the higher order coefficients diminish, Equations (14), (15) and (18) indicate that the free boundary will tend to become an ellipse.

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Zusammenfassung Es wird die Verformung einer zylindrischen freien Oberfläche in einem unendlichen viskosen Körper untersucht, der sich unter dem Einfluss eines gleichförmigen Spannungszustandes im Unendlichen in einem ebenen Fliesszustand befindet. Es wird gezeigt, dass dann, wenn die freie Oberfläche ursprünglich durch ein Polynomn-ten Grades konform auf einen Kreis abgebildet werden kann, das gleiche auch für die deformierte Oberfläche gilt. Für diese Problemklasse wird eine Methode zur Ermittlung der exakten Oberflächenbewegung angegeben und an einem Beispiel illustriert. Dabei zeigt sich, dass eine kleine Unregelmässigkeit in der freien Oberfläche unter Druck im Unendlichen anwachsen kann.

     

A note on nonlinear stability of plane parallel shear flows

We present a generalized energy functional E for plane parallel shear flows which provides conditional nonlinear stability for Reynolds numbers Re below some value ReE depending on the shear profile. In the case of the experimentally important profiles, viz. combinations of laminar Couette and Poiseuille flow, ReE is shown to be at least 174.     

A note on plane Stokes flow past a shear free impermeable cylinder

Summary The slow steady two-dimensional motion of a viscous incompressible fluid in the unbounded region exterior to a shear free circular cylinder which is impermeable is examined. It is shown that the above problem requires a certain consistency condition for the existence of a solution. In addition, a circle theorem for the biharmonic equation is presented, for the above plane Stokes flow. Some examples are also given.     

Problems with free boundaries for nonlinear parabolic equations

We establish necessary conditions for the existence of effects of space localization and stabilization in time that are qualitatively new for evolutionary equations. We suggest constructive methods for the solution of the corresponding one-dimensional problems with free boundaries that appear in ecology and medicine. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1360–1372, October, 1997.     

A note on the regularity of the free boundaries in the optimal partial transport problem

In this work we study the global regularity of the free boundaries arising in the optimal partial transport problem. Assuming the supports of both the source and the target measure to be convex, we show that the free boundaries of the active regions are globally C ^0,1/2.      

A viral propagation model with a nonlinear infection rate and free boundaries

In this paper we put forward a viral propagation model with a nonlinear infection rate and free boundaries and investigate the dynamical properties. This model is composed of two ordinary differential equations and one partial differential equation, in which the spatial range of the first equation is the whole space R, and the last two equations have free boundaries. As a new mathematical model, we prove the existence,uniqueness and uniform estimates of the global solution, and provide the criteria for spreading and vanishing, and the long time behavior of the solution components u, v and w. Comparing this model with the corresponding ordinary differential systems, the basic reproduction number R_0 plays a different role. We find that when R_0≤1, the virus cannot spread successfully; when R_0 1, the successful spread of the virus depends on the initial value and varying parameters.     

A note on plane pointless curves

Let $d(q)$ denote the minimal degree of a smooth projective plane curve that is defined over the finite field ${\mathbb{F}}_q$ and does not contain ${\mathbb{F}}_q$ rational points. We are interested in the asymptotic behavior of $d(q)$ for $q\to \infty$. To the best of the author's knowledge the problem of estimating the asymptotic behavior of $d(q)$ was not considered previously. In this note we establish the following bounds:(1)$\frac{1}{4}⩽\underset{q\to \infty }{\underline{\lim }}{\log }_{q}d(q)⩽\frac{1}{3}.$ More specifically, for every characteristic $p>3$ we construct a sequence of pointless Fermat curves$x^{d_k}+y^{d_k}+z^{d_k}=0,\text{over }{\mathbb{F}}_{p^{m_k}},$ such that ${\lim }_{k\to \infty }{\log }_{p^{m_k}}{d}_k=1/3$.     

Fully nonlinear singularly perturbed equations and asymptotic free boundaries

In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζε(u) with ζε→δ₀⋅∫ζ. We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F(D²u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.     

A note on pseudo-metrics on the plane

Every c-finite measure Μ on the set G of the lines on the plane such that $$(0){\text{ }}\mu {\text{(\{ g}} \in G:{\text{ }}P \in {\text{g\} ) = 0}}$$ for every point P?R ² generates a pseudo-metric F on the plane when one puts F P ₁, P ₂= \(\tfrac{1}{2}\) μ({g∈G:g separates the points P ₁ and P ₂}) The pseudo-metrics which are generated in this way possess the property of linear additivity, that is F(P ₁,P ₃)=F(P ₁,P ₂)+F(P ₂,P ₃) for P ₁,P ₂,P ₃ on a line, P ₂ between P ₁ and P ₃, and are continuous with respect to the Euclidean topology in R ² × R ². In this paper we prove the converse: every linear additive and continuous pseudo-metric F is generated as above by some c-finite measure Μ on G for which (0) holds. The method of proof shows that values of linearly additive and continuous pseudo-metric F inside every bounded convex polygon C are determined completely by the values of F on (δC)². The representation of pseudo-metrics by measures is useful in derivation of inequalities for the former.     

A note on coverings of plane graphs

For any plane graph G the number of edges in a minimum edge covering of the faces of G is at most the vertex independence number of G and the numbre of vertices in a minimum vertex covering of the faces of G is at most the edge independence number of G. © 1995 John Wiley & Sons, Inc.     

Problems with free boundaries and nonlocal problems for nonlinear parabolic equations

We present statements of problems with free boundaries and nonlocal problems for nonlinear parabolic equations arising in metallurgy, medicine, and ecology. We consider some constructive methods for their solution. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 84–97, September, 1996.     

A note on tropical triangles in the plane

We define transversal tropical triangles （affine and projective） and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.     

A note on list improper coloring of plane graphs

A list-assignment L to the vertices of G is an assignment of a set L(v) of colors to vertex v for every v∈V(G). An (L,d)^∗-coloring is a mapping ? that assigns a color ?(v)∈L(v) to each vertex v∈V(G) such that at most d neighbors of v receive color ?(v). A graph is called (k,d)^∗-choosable, if G admits an (L,d)^∗-coloring for every list assignment L with |L(v)|≥k for all v∈V(G). In this note, it is proved that every plane graph, which contains no 4-cycles and l-cycles for some l∈{8,9}, is (3,1)^∗-choosable.     

A note on generalized nonlinear diagonal dominance

In this paper, an open problem, proposed by A. Frommer, about nonlinear generalized diagonal dominance, is solved on some weak restriction, a counterexample is presented if such a restriction is omitted, and some new properties of nonlinear generalized diagonally dominant functions are investigated.     

A note on intersections of free submonoids of a free monoid

According to a theorem of Tilson [6] any intersection of free submonoids of a free monoid is free. Here we consider intersections of the form {x, y}^* ∩ {u, v}^*, where x, y, u and v are words in a finitely generated free monoid Σ^*, and show that if both the monoids {x, y}^* and {u, v}^* are of the rank two, then the intersection is a free monoid generated either by (at most) two words or by a regular language of the form β₀ + β((γ(1+ δ + ... δ^t))^*ε for some words β⁰, β, γ, δ and ε, and some integer t≥0. An example is given showing that the latter possibility may occur for each t≥0 with nonempty values of the words.