Let C2+, (0, 1), then for small time t [0, t0] the surface (t) may be represented by the equation [1], [2] x = + (, t) N(), = (x), t [0, t0], where t=0 = 0, N() = (N1,..., Nn) is a unit vector field determined on and such that N() C2+(; Rn), 0() NT () d1 =const> 0, 0() unit normal to, NT vector-row, 0() NT () scalar product.

Theorem 1 Let C2+, (0, 1). For every function u0(x, t) C2+() satisfying the compatibility conditions u0 = 0, |u0(x)| = 1, and the condition | u0| d2, d2 = const > 2+,1+/ 0, there exists T0 > 0 such that the problem (1)–(3) has a unique solution u(x, t) Cxt (QT ), 2+,1+/(, t) Cxt (T ) and an estimate for it is fulfilled |u|(2+) + ||(2+) c|u0|(2+), t [0, T0].

Qt t Here c is a positive constant.

References 1. Bizhanova G.I., Solonnikov V.A. On the free boundary problems for the parabolic equations.

Algebra i Analiz. 2000. V.12, №6. P.3–45.

2. Hanzawa E.I. Classical solutions of the Stefan problem. Tohoku Math. J. 1981. V.33, №3. P.

297–335.

Chemetov N. V. GENERALIZED KAUP KUPERSHMIDT SOLUTION, STATIC SOLUTIONS AND BOUNDARY GENERATED SOLUTIONS SOLUTIONS OF THE KdV AND HIGH ORDER KdV EQUATIONS I. Burde Georgy Jacob Blaustein Institutes for Desert Research, Ben-Gurion University, Israel The famous Korteweg de Vries (KdV) equation discovered back in 1895 for waves on water arises in many physical contexts as an equation governing weakly nonlinear long waves when nonlinearity and dispersion are in balance at leading order. If higher order nonlinear and dispersive effects are of interest, then the asymptotic expansion can be extended to the higher orders in the wave amplitude which leads to the higher-order KdV equations. Some of those equations are generic, i.e. they can be derived, using physically meaningful asymptotic techniques, from very large classes of nonlinear PDE’s. The importance of exact solutions of the equations derived in the realm of perturbation methods comes from the fact that they may describe asymptotic limits of solutions of equations supposed to govern real systems.

In this paper, new types of exact and explicit solitary wave solutions of the KdV and higherorder KdV equations are found using a direct method, designed specifically for constructing solitary wave solutions of evolution equations. The first type is the ’generalized Kaup Kupershmidt’ (GKK) solitary waves [1], which unify the structures of the sech2 KdV-like soliton and the Kaup Kupershmidt soliton and also provide solutions of some other equations. One of those equations is found to possess the multi-soliton solutions which makes it a good candidate for an equation integrable in terms of the GKK solitons. Another type of solutions expressed in terms of algebraic functions represents the steady-state localized structures which may be considered as (static) solitons. This view is based on the solutions describing interaction of those steady-state localized patterns with regular solitons. It is seen that the steady-state structures behave as solitons when they collide with regular solitons their shape remains unchanged after the collision, only a phase (coordinate) shift is observed. Note that such solutions containing both hyperbolic and algebraic functions cannot be obtained by applying the popular “tanh”, “sinh” and so on methods. Some of the solutions obtained can be applied to the initial-boundary-value problem which arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg de Vries equation. Such a ’boundary generated’ solitons (solutions of a ’quarter-plane problem’) have received attention in the past being treated numerically or asymptotically. In the present paper, solutions both of the standard KdV equation and of the higher-order KdV equations, which describe solitons propagating from the boundary at which a constant value of the variable is maintained, are defined.

References 1. Burde G. I. J. Phys. A: Math. Theor. 43 (2010) 085208 (13pp).

BOUNDARY LAYER PROBLEM: NAVIER- AND EULER EQUATIONS N. V. Chemetov Centro de Matemtica e Aplicaes Fundamentais, Lisbon We consider the Navier Stokes equations in a 2D-bounded domain Rvt + div (v v) - p = µv, x, t > 0, Chemetov N. V. div v = 0, v(x, 0) = v0(x), x, admitting flows through the boundary of v · n = a(x, t), x, t > 0, 2D(v)n · s + (x, t)v · s = b(x, t).

The last one are so-called Navier slip boundary conditions. Here v(x, t) the velocity of the fluid;

p(x, t) the pressure; D(v) = [v + (v)T ] the rate-of-strain tensor of v; (n, s) the pair formed by the outside normal and tangent vectors to the boundary of.

The main result of our work: Under the viscous parameter µ 0 we shown that the solutions vµ of the Navier Stokes equations converge to the solution v of the Euler equations, satisfying the Navier slip boundary conditions on the part of the boundary where v · n = a < 0, such that vµ v strongly in L(0, T ; Wp ()).

This result solved a so-called problem of boundary layers.

References 1. Chemetov N. V., Starovoitov V. N. On a Motion of a Perfect Fluid in a Domain with Sources and Sinks J. Math. Fluid Mechanics, 4, No. 2, 128–144 (2002).

2. Chemetov N. V., Antontsev S. N. Euler equations with non-homogeneous Navier slip boundary condition Physica D: Nonlinear Phenomena, 237, 1, 92–105 (2008).

3. Chemetov N. V., Cipriano F., Gavrilyuk S. Shallow water model for lakes with friction and penetration Mathematical Methods in the Applied Sciences, published online, (2009).

NONLINEAR HYPERBOLIC – ELLIPTIC SYSTEMS N. V. Chemetov Centro de Matemtica e Aplicaes Fundamentais, Lisbon We investigate a mixed hyperbolic-elliptic type system of PDEs in a bounded domain RN t + div (g()v) = 0 with v = -h -h + h =.

which is closed by the natural condition on the boundary of h = a(x, t) and the initial condition |t=0 = 0(x) in.

Motivated by physics, on the influx part of, i.e. on -(t) := {(x, t) : g (v · n)(x, t)) < 0}, Christov C. I. we consider nonzero boundary condition h = b(x, t, ).

n Here n is the outside normal to.

We prove the solvability of this system, using a kinetic formulation of the problem. The system can be used for different physical situations, such as a) the motion of superconducting vortices in the superconductor;

b) the collective cell movement (the Keller Segel model).

References 1. Antontsev S. N., Chemetov N. V. “Flux of superconducting vortices through a domain SIAM Journal on Mathematical Analysis, 39, No. 1, 263–280 (2008).

2. Antontsev S. N., Chemetov N. V. Superconducting Vortices: Chapman Full ModelNew Directions in Mathematical Fluid Mechanics, Verlag Basel/Switzerland, Editors: Fursikov A. V., Galdi G., Pukhnachev V. V., 41–55 (2010).

NONLINEAR CONTINUUM MECHANICS OF SPACE AND THE FRAME-INDIFFERENT (TRULY COVARIANT) FORMULATION OF ELECTROMAGNETISM C. I. Christov Department of Mathematics, University of Louisiana at Lafayette, USA We prove that, when linearized, the governing equations of an incompressible elastic liquid yield Maxwell’s equations as corollaries. The divergence of the deviator stress tensor is interpreted as the electric field, while the vorticity (the curl of velocity field) is interpreted as the magnetic field. Thus we establish that the electrodynamics can be fully explained as the manifestation of the internal stresses of a material space which we term the metacontinuum. Through judicious distinction between the referential (Lagrange) and local (Euler) descriptions, the principle of material invariance (frame indifference) is established and shown to be a true covariance principle, unlike the Lorentz covariance, which is valid only for non-deforming frames in rectilinear relative motion. The new nonlinear formulation of the electrodynamics incorporates the Lorentz force as an integral part of Faraday’s law, rather than as an additional empirical observable relation. We show that the Ampere-Oersted and Biot Savart laws can be derived from the frame-indifferent modification of Maxwell’s displacement current. The material invariance of the model entails Galilean invariance as a limiting case.

We propose a new concept in which the particles and charges are considered as localized nonlinear waves (solitons). They do not move through, but rather propagate over the surface of the metacontinuum. We show that a localized screw deformation possesses a topological charge and for all practical purposes can be interpreted as the charge. It is a well established fact in soliton theory that a propagating phase pattern is contracted in the direction of propagation by the Lorentz factor.

This means that the Lorentz contraction is an intrinsic property of the proposed model without Lorentz Transformation. We address also the conundrum connected with the detectability of the absolute continuum. The phase-change concept of Michelson interferometer is unable, in principle, to detect the first-order Doppler effect, while the Lorentz contraction cancels exactly the secondorder. We reexamine the seminal experiment of Ives and Stilwell using a modified Bohr Rydberg Finn R. formula accounting for the motion of the emitting atom. We show that the results of Ives and Stilwell are fully compatible with the presence of an absolute medium, without assuming time dilation. Finally, we propose a new interferometry concept based on the beat frequency, which can produce results for the first-order Doppler effect, allowing detecting the relative speed with respect to the preferred frame.

The model of space as an elastic continuum with relaxation (elastic Maxwell liquid) allows a reformulation of the electromagnetism in a truly covariant form: the new equations yield in the linear limit Maxwell’s equation, while their full nonlinear version is frame indifferent. Thus the new formulation is invariant in any non-inertial frame that can also deform during the motion.

References 1. Christov C.I., On the Nonlinear Continuum Mechanics of Space and the Notion of Luminiferous Medium, Nonlinear Analysis, 71 (2009), e2028–e2044.

FLOATING BODIES ON CAPILLARY INTERFACES R. Finn Stanford University, USA In classical capillarity theory one assumes a rigid ”support” surface (e.g., capillary tube) that is fixed in space, and one seeks to describe the surface of an adjacent liquid (into which one dips the tube), in accordance with governing physical laws. For the related problem of a body floating at a fluid interface endowed with surface tension, the relevant physical laws are the same, however neither the liquid surface nor the rigid support is at a fixed position in space, and each is subject to different kinds of constraints. Thus a new level of subtlety appears, and new procedures are needed to obtain useful information. The present work offers an initial step toward characterizing the configurations that can occur, in accordance with classical energy principles.

Several specific problems are addressed, notably that of determining conditions under which a body whose density exceeds that of the ambient liquid will float or sink. In a downward gravity field, floating configurations yield in general local energy minima that are not global, as the energy can be made negatively unbounded by submerging the body to increasing depth.

Both the (2-d) two dimensional case (an infinite cylinder floating horizontally) and also the 3-d case of a compact body are considered, in both cases under a convexity hypothesis. There are distinct differences in behavior. In 2-d the height h of the region |z| < h surrounding the undisturbed interface z = 0 is explicitly characterized, such that no floating cylinder can be disjoint from that set. This result is independent of cylinder shape or size, or of its composition. In 3-d one finds again such a (universal) set, however smaller objects are now shown to be restricted to regions closer to the level surface, reflecting physical experience. Again this result is explicit.

In all cases it is shown that for suitable surface energy densities, floating configurations providing local energy minima that are not global can always be realized. In 3-d, surface energy relations are characterized under which any body, of any density, can be made to float by scaling it to smaller size.

In general, bodies will not float freely in every orientation. Nevertheless, in 2-d for contact angle /2 there is an infinity of distinct sectional shapes for which the cylinder can achieve floating equilibrium regardless of orientation. Representative such shapes are obtained explicitly. Noncircular sections with this property exist for at most a countable infinity of contact angles.

The analogous question in 3-d leads to the quite different conclusion, that regardless of contact angle the only closed surface with such a property is a metric sphere. In both the 2-d and 3-d cases, Grebenev V. N. it is shown that if the body is moved rigidly into the fluid from above, or out of the fluid from below, in any fixed orientation, the ambient fluid interface must change configuration discontinuously.

Some of this work was joint with Mattie Sloss, and some was joint with Thomas Vogel.

ON THE GEOMETRY OF THE CORRELATION SPACE FOR ISOTROPIC TURBULENCE V. N. Grebenev Institute of Computational Technologies SB RAS, Novosibirsk A new geometric view of homogeneous isotropic turbulence is contemplated employing the two-point velocity correlation tensor of the velocity fluctuations. We show that this correlation tensor generates a family of the so-called semi-reducible pseudo-Riemannian metrics. This enables us to specify the geometry of a singled out Eulerian fluid volume in a statistical sense and to introduce into the consideration the structure of a semi-reducible pseudo-Riemannian manifold on the correlation space. We expose the relationship of some geometric constructions with statistical quantities arising in turbulence. In particular, the formula that connects the Gaussian curvature K of the model manifold M constructed (which is realized as a surface of revolution) and the Taylor microscales 2 and 2 is given. Moreover, using the well-known relationship between turbulent f(tc) c g(t ) length scales g(t ) = 102/3L1/3 where ( = (3/)1/4) is the Kolmogorov length scale, L is the c integral length scale characterizing the large eddies, denotes the viscosity and is the dissipation of turbulent energy, we establish that in the limit of infinite Reynolds numbers or vanishing the viscosity, the Gaussian curvature K of the cross-section of the model manifold M grows infinitely.

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