Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation in (0.1) $ u_t = (uu_x )_{xx} in\mathbb{R} \times \mathbb{R}_ + . $ It is shown that two basic Riemann problems for Eq. (0.1) with the initial data $ S_ \mp (x) = \mp \operatorname{sgn} x $ exhibit a shock wave (u(x, t) ≡ S _?(x)) and a smooth rarefaction wave (for S ₊), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u _t + uu _x = 0, which was developed by O.A. Oleinik and S.N. Kruzhkov (for equations in x ? ? ^N ) in the 1950s–1960s.     

The Cauchy Problem for a Tenth-Order Thin Film Equation I. Bifurcation of Oscillatory Fundamental Solutions

Fundamental global similarity solutions of the tenth-order thin film equation $$u_{t} = \nabla . (|u|^{n} \nabla \Delta^{4}u) \,\,\,\, {\rm in} \,\,\,\, \mathbb{R}^{N} \times \mathbb{R}_{+}$$ , where n >  0 are studied. The main approach consists in passing to the limit ${n \rightarrow 0^{+}}$ by using Hermitian non-self-adjoint spectral theory corresponding to the rescaled linear poly-harmonic equation $$u_{t} = \Delta^{5}u \,\,\,\, {\rm in} \,\,\,\, \mathbb{R}^{N} \times \mathbb{R}_{+}$$ .     

On nonexistence of Baras-Goldstein type without positivity assumptions for singular linear and nonlinear parabolic equations

The celebrated result by Baras and Goldstein (1984) established that the heat equation with the inverse square potential in the unit ball B ₁ ⊂ ℝ ^N , N ≥ 3, u _t = Δ _u + in B ₁ × (0,T), u|∂B ₁ = 0, in the supercritical range c > c _Hardy = does not have a solution for any nontrivial L ¹ initial data u ₀(x) ≥ 0 in B ₁ (or for a positive measure u ₀). More precisely, it was proved that a regular approximation of a possible solution by a sequence {u _n (x,t)} of classical solutions corresponding to truncated bounded potentials given by V(x) = ↦ V _n (x) = min{, n} (n ≥ 1) diverges; i.e., as n → ∞, u _n (x,t) → + ∞ in B ₁ × (0, T). Similar features of “nonexistence via approximation” for semilinear heat PDEs were inherent in related results by Brezis-Friedman (1983) and Baras-Cohen (1987). The main goal of this paper is to justify that this nonexistence result has wider nature and remains true without the positivity assumption on data u ₀(x) that are assumed to be regular and positive at x = 0. Moreover, nonexistence as the impossibility of regular approximations of solutions is true for a wide class of singular nonlinear parabolic problems as well as for higher order PDEs including, e.g., u _t = , and , N > 4. Dedicated to Professor S.I. Pohozaev on the occasion of his 70th birthday     

Nonlinear dispersion equations: Smooth deformations,compactions, and extensions to higher orders

The third-order nonlinear dispersion PDE, as the key model,

Calculation of thermoelastic stresses near the crystallization front for melt-grown single-crystal rods with a circular cross section

Approximate expressions for the thermoelectric stress tensor components in an isotropic crystal rod of a circular cross section, applicable in the entire crystal including the region near the crystallization front, are obtained. Using the resultant approximate formulas, the stress fields in leucosapphire single crystals are calculated for model temperature fields. It is shown that exactly near the crystallization front, thermoelastic stresses reach maximal values.     

Preliminary results of an aircraft system based on near-IR diode lasers for continuous measurements of the concentration of methane, carbon dioxide, water and its isotopes

The Federal Agency for Hydrometeorology of the Russian Federation created the flying laboratory on board the passenger airplane Yak-42D for geophysical monitoring of the environment, including aircraft measurements of vertical concentrations of greenhouse gases in the troposphere. Within the limits of this project, General Physics Institute of the Russian Academy of Science developed airborne tunable diode laser spectrometer (TDLS) on the basis of diode lasers of a near-IR range for measurement of the altitude profiles of CO₂, CH₄, H₂O and its isotopes. TDLS complex was integrated aboard in standard 19-in. rack. Air samples, taken over an aircraft on the pipeline, were injected into the optical cell. Using the system of inflow and heating, the air was set laminar with a flowrate of 0.2?l/s at a reduced pressure of 100?mbar for detecting narrow absorption lines of water vapor isotopes. For registration of the absorption spectra and for the measurement of greenhouse gas concentrations in online mode, modulation-correlation technique was used. Diode laser spectrometer output data were transferred to the airborne central computer. Sensitivity of TDLS measurements was 20?C30?ppm for water, 3?C4?ppm for CO₂ and 20?C25?ppb for CH₄. Time of one-unit measurement is about 30?ms.     

On the absence of global solutions of nonlinear mixed problems

The paper gives examples demonstrating the blow-up phenomenon for solutions of partial differential equations. It is shown that the nature of blow-up of solutions of the mixed problem substantially differs from that of solutions of the Cauchy problem. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 71–89, 2007.