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The main concern of this paper is to mathematically investigate the formation of a plasma sheath near the surface of nonplanar walls. We study the existence and asymptotic stability of stationary solutions for the nonisentropic…

We show possibility of the Plane Couette (PC) flow instability for Reynolds number Re>Reth=140. This new result of the linear hydrodynamic stability theory is obtained on the base of refusal from the traditionally used assumption on…

In this paper, Neumann cracks in elastic bodies are considered. We establish a rigorous asymptotic expansion for the boundary perturbations of the displacement (and traction) vectors that are due to the presence of a small elastic linear…

Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by…

Nonlinear partial differential equations are central to physics, engineering, and finance. Except in a limited number of integrable cases, their solution generally requires numerical methods whose cost becomes prohibitive in…

This paper is concerned with the stability of steady solutions to initial-boundary-value problems of reaction-hyperbolic systems for axonal transport. Under proper structural assumptions, we clarify the relaxation structure of the…

A general system of n ordinary differential equations coupled with one reaction-diffusion equation, considered in a bounded N-dimensional domain, with no-flux boundary condition is studied in a context of pattern formation. Such initial…

This paper develops a new approach to the estimation of the degree of boundedness or stability of multidimensional nonlinear systems with time-dependent nonperiodic coefficients-an essential task in various engineering and natural science…

The linear evolution of disturbances due to a ribbon vibrating at frequency $\omega_0$ in plane Poiseuille flow is computed by solving the associated initial boundary value problem in the Fourier-Laplace plane, followed by inversion. A…

The dynamics and stability of a fluid-filled hollow cylindrical shell rolling on an inclined plane are analyzed. We study the motion in two dimensions by analyzing the interaction between the fluid and the cylindrical shell. An analytical…

We investigate the evolution of the surface of radiating stars by studying the asymptotic behaviour of exact solutions initiated via the stationary boundary condition. This boundary condition leads to a master equation in the form of a…

In this paper, we derive sharp asymptotics for the spectral data (eigenvalues and weight numbers) of the fourth-order linear differential equation with a distribution coefficient and three types of separated boundary conditions. Our methods…

This paper is on the effect of nonlinearity in the equations for propagation of disturbances on transition in the class of Spiral Poiseuille Flows. The problem is approached from the fundamental point of view of following the growth of…

This paper deals with the exponential stability of systems made of a hyperbolic PDE coupled with an ODE with different time scales, the dynamics of the PDE being much faster than that of the ODE. Such a difference of time scales is modeled…

In this paper, we propose a numerical framework to study the shapes, dynamics and the stabilities of the self-localized solutions of the nonlinear wave blocking problem. With this motivation, we use the nonlinear Schr\"odinger equation…

We study the linear dynamics of spectrally stable $T$-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr\"odinger equation with forcing that arises in nonlinear optics. Such $T$-periodic solutions…

The spatio-temporal dynamics of separation bubbles induced to form in a fully-developed turbulent boundary layer (with Reynolds number based on momentum thickness of the boundary layer of 490) over a flat plate are studied via direct…

At sufficiently high Reynolds numbers, shear-flow turbulence close to a wall acquires universal properties. When length and velocity are rescaled by appropriate characteristic scales of the turbulent flow and thereby measured in \emph{inner…

By a refinement of the technique used by Johnson and Zumbrun to show stability under localized perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction…

Analytical solutions in fluid dynamics can be used to elucidate the physics of complex flows and to serve as test cases for numerical models. In this work, we present the analytical solution for the acoustic boundary layer that develops…