Since the late 1990s, goal-oriented error estimation and goal-oriented adaptive methods have been developed to control the discretization error in goal functionals of the solution1,2. These methods have mostly been applied to linear and nonlinear problems in solid and fluid mechanics. An important recent development is the extension of goal-oriented adaptive methods to multiphysics problems inv...

For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows, this approach has failed to match experimental results. Recently, new scenarios for transition have been proposed that are based on the non-normality of the linearized operator. These new “mostly li...

We consider the logarithmic Korteweg–de Vries (log–KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in H(R) with conserved L norm and energy, we construct a weak global solution of the log–KdV equation in a subset of H(R). This construction yields c...

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $$L_K(\mathcal {G})$$ of an ultragraph $$\mathcal {G}$$ over a field K is purely infinite simple that it von Neumann regular. Consequently, obtain every graded either locally matricial algebra, or full matrix ring $$K[x, x^{-1}]$$ , algebra.

Metastable dynamics, which qualitatively refers to physical processes that involve an extremely slow approach to their nal equilibrium states, is often associated with singularly perturbed convection-di usion-reaction equations. A problem exhibits metastable behavior when the approach to equilibrium occurs on a time-scale of order O(eC" ), where C > 0 and " is the singular perturbation paramete...

We prove that the essential spectrum of the operator obtained by linearization about a steady state of the Euler equations governing the motion of inviscid ideal fluid in dimension two is a vertical strip whose width is determined by the maximal Lyapunov exponent of the flow induced by the steady state.

This paper is concerned with an integral equation that models discrete time dynamics of a population in patchy landscape. The patches the domain are reflected through discontinuity kernel operator at finite number points whole domain. We prove existence and uniqueness stationary state under certain assumptions on principal eigenvalue linearized growth term as well. also derive criteria which un...

In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg–Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feed...