On the structure of the set of continuously differentiable solutions of one boundary-value problem for a system of functional differential equations of neutral type

We study the structure of the set of solutions, continuously differentiable for t ∈ R ⁺ = [0; + ∞), of one limit problem for systems of nonlinear functional differential equations of neutral type with nonlinear deviations of argument that depend on an unknown function. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 277–289, April–June, 2007.     

Temporal and Spatial Decays for the Stokes Flow

We estimate the time decay rates in L ¹, in the Hardy space and in L ^∞ of the gradient of solutions for the Stokes equations on the half spaces. For the estimates in the Hardy space we adopt the ideas in [7], and also use the heat kernel and the solution formula for the Stokes equations. We also estimate the temporal-spatial asymptotic estimates in L ^q , 1 < q < ∞, for the Stokes solutions. This work was supported by grant No. (R05-2002-000-00002-0(2002)) from the Basic Research Program of the Korea Science & Engineering Foundation.     

Analyticity and Decay Estimates of the Navier–Stokes Equations in Critical Besov Spaces

In this paper, we establish analyticity of the Navier–Stokes equations with small data in critical Besov spaces . The main method is Gevrey estimates, the choice of which is motivated by the work of Foias and Temam (Contemp Math 208:151–180, 1997). We show that mild solutions are Gevrey regular, that is, the energy bound holds in , globally in time for p < ∞. We extend these results for the intricate limiting case p = ∞ in a suitably designed E _∞ space. As a consequence of analyticity, we obtain decay estimates of weak solutions in Besov spaces. Finally, we provide a regularity criterion in Besov spaces.     

On the asymptotic properties of solutions of linear functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation {fx153-01} in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 147–150, April–June, 2008.     

The Fundamental Solution of the Linearized Navier–Stokes Equations for Spinning Bodies in Three Spatial Dimensions – Time Dependent Case

Explicit formulae for the fundamental solution of the linearized time dependent Navier–Stokes equations in three spatial dimensions are obtained. The linear equations considered in this paper include those used to model rigid bodies that are translating and rotating at a constant velocity. Estimates extending those obtained by Solonnikov in [23] for the fundamental solution of the time dependent Stokes equations, corresponding to zero translational and angular velocity, are established. Existence and uniqueness of solutions of these linearized problems is obtained for a class of functions that includes the classical Lebesgue spaces L^p(R³), 1 < p < ∞. Finally, the asymptotic behavior and semigroup properties of the fundamental solution are established.     

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ω_ε that is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H ¹(Ω_ε) as ε → 0 (N → +∞). Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006.     

Laminar-turbulent transition in the boundary layer on cones in a hypersonic flow at high Reynolds numbers per meter

The laminar-turbulent transition is experimentally studied in boundary-layer flows on cones with a rectangular axisymmetric step in the base part of the cone and without the step. The experiments are performed in an A-1 two-step piston-driven gas-dynamic facility with adiabatic compression of the working gas with Mach numbers at the nozzle exit M _∞ = 12–14 and pressures in the settling chamber P₀ = 60–600 MPa. These values of parameters allow obtaining Reynolds numbers per meter near the cone surface equal to Re _1e = (53–200) · 10⁶ m ⁻¹. The transition occurs at Reynolds numbers Re _tr = (2.3–5.7) · 10⁶. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 76–83, May–June, 2007.     

Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in {\mathbb {R}^3}

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}The compressible Navier–Stokes–Poisson (NSP) system is considered in \mathbb R³{\mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate (1+t)^{-\frac 14}{(1+t)^{-\frac {1}{4}}} or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with p ? (1, 3/2){p \in (1, 3/2)} for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.     

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L ^∞ in space and time.     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

Convergence Rate for Compressible Euler Equations with Damping and Vacuum

 We study the asymptotic behavior of L ^∞ weak-entropy solutions to the compressible Euler equations with damping and vacuum. Previous works on this topic are mainly concerned with the case away from the vacuum and small initial data. In the present paper, we prove that the entropy-weak solution strongly converges to the similarity solution of the porous media equations in L ^p (R) (2≤p<∞) with decay rates. The initial data can contain vacuum and can be arbitrary large. A new approach is introduced to control the singularity near vacuum for the desired estimates. (Accepted August 31, 2002) Published online January 9, 2003 Communicated by C. M. Dafermos     

On asymptotic properties of solutions of linear differential functional equations with linearly transformed argument

We establish new properties of C ¹[−1, +∞)-solutions of the linear functional differential equation ẋ(t) = ax(t) + bx(qt) + hx(t−1) + cẋ(qt) + rẋ(t−1) in the neighborhood of the singular point t = +∞. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 170–177, April–June, 2006.     

Certain features of similarity solutions for flows in viscous vortex cores

A class of steady similarity solutions of the equations for viscous vortex cores which correspond to external inviscid similarity solutions with a power-law variation of the circumferential velocityv-r ^−m near the rotation axis is considered. It is found that if the Bernoulli function in external flow is constant, then these solutions will exist only on a certain range of the indexm of the exponential. For eachm on this range there are two solutions. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–43, January–February, 1998. The work was financially supported by the Russian Foundation for Fundamental Research (project No. 95-01-00483).     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

Experimental study and direct numerical simulation of the evolution of disturbances in a viscous shock layer on a flat plate

The evolution of disturbances in a hypersonic viscous shock layer on a flat plate excited by slow-mode acoustic waves is considered numerically and experimentally. The parameters measured in the experiments performed with a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵ are the transverse profiles of the mean density and Mach number, the spectra of density fluctuations, and growth rates of natural disturbances. Direct numerical simulation of propagation of disturbances is performed by solving the Navier-Stokes equations with a high-order shock-capturing scheme. The numerical and experimental data characterizing the mean flow field, intensity of density fluctuations, and their growth rates are found to be in good agreement. Possible mechanisms of disturbance generation and evolution in the shock layer at hypersonic velocities are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 3–15, September–October, 2006.     

Stability for the 3D Navier–Stokes Equations with Nonzero far Field Velocity on Exterior Domains

Concerning to the non-stationary Navier–Stokes flow with a nonzero constant velocity at infinity, just a few results have been obtained, while most of the results are for the flow with the zero velocity at infinity. The temporal stability of stationary solutions for the Navier–Stokes flow with a nonzero constant velocity at infinity has been studied by Enomoto and Shibata (J Math Fluid Mech 7:339–367, 2005), in L ^p spaces for p ≥ 3. In this article, we first extend their result to the case \frac32 < p{\frac{3}{2} < p} by modifying the method in Bae and Jin (J Math Fluid Mech 10:423–433, 2008) that was used to obtain weighted estimates for the Navier–Stokes flow with the zero velocity at infinity. Then, by using our generalized temporal estimates we obtain the weighted stability of stationary solutions for the Navier–Stokes flow with a nonzero velocity at infinity.     

Self-sustained oscillations and bifurcations of transonic flow past simple airfoils

A turbulent flow past two symmetric airfoils, whose bow and aft portions are circular arcs, whereas midparts are flat, is studies numerically. The amplitude of lift coefficient oscillations versus the free-stream Mach number M _∞ is analyzed at zero angle of attack. Ranges of M _∞ in which there exist flow bifurcations are determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 37–44, November–December, 2008     

Properties of similarity solutions for flows in turbulent vortex cores

A class of similarity solutions of the equations for turbulent vortex cores matching an external inviscid similarity flow with a power law of circumferential velocity variationv-r ^−m near the rotation axis and constant Bernoulli function is considered. Solutions are found to exist only in a certain range of the indexm of the exponential. For each suchm there are two solutions. The authors wish to apologise for a mistake which resulted in the figures in this paper corresponding to [1] and those in [1] corresponding to this paper. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 60–64, May–June, 1998. The work was financially supported by the Russian Foundation for Basic Research (project No. 95-01-00483).     

Decay and Continuity of the Boltzmann Equation in Bounded Domains

Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ^∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L ² decay theory and its interplay with delicate L ^∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.