In this paper, we study the motion of spirals by mean curvature in a (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line attains the surface of a crystal. The main result of this paper is a comparison principle for the corresponding quasi-linear equation. As far as motion of spirals are concerned, the novelty and...

The rigorous Method of Regularization is applied to the problem of electromagnetic scattering of plane waves by a finite array of partiallyshielded circular cylinders. A set of ill-posed dual series equations is resulted from imposing the mixed boundary conditions on the surface of each of the cylinders. Each of the M sets of dual series equations are treated separately using the Method of Regu...

We prove that the KdV-Burgers is globally well-posed in H−1(T) with a solution-map that is analytic fromH−1(T) to C([0, T ];H−1(T)) whereas it is ill-posed in Hs(T), as soon as s < −1, in the sense that the flow-map u0 7→ u(t) cannot be continuous from H s(T) to even D′(T) at any fixed t > 0 small enough. In view of the result of Kappeler and Topalov for KdV it thus appears that even if the dis...

Abstract. This note concerns a nonlinear ill-posedness of the Prandtl equation and an invalidity of asymptotic boundary-layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear ill-posedness results established by Gérard-Varet and Dormy [2], and an analysis in Guo and Tice [5]. We show that the asymptotic boundary-layer expansion ...

In this paper, we modify the adaptive wavelet algorithm from [Technical Report 1325, Department of Mathematics, Utrecht University, March 2005] so that it applies directly, i.e., without forming the normal equation, not only to self-adjoint elliptic operators but also to such operators to which generally nonsymmetric lower order terms are added, assuming that the resulting operator equation is ...

Our aim in this paper is to investigate the global attractivity of all positive solutions of the higher order nonlinear difference equation xn+1 = a+ bxn A−∑k i=0 bixn−i , n = 0, 1, . . . , where A, b, bk ∈ (0,∞), a, b0, . . . , bk−1 ∈ [0,∞), k ∈ {1, 2, . . .} and the initial conditions x−k, . . . , x0 are arbitrary. We show that one positive equilibrium of the equation is a global attractor wi...

In this paper, we study a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain T (a unit circle in the plane) with an internal control acting on an arbitrary small nonempty subdomain of T. Aided by the Bourgain smoothing property of the Kawahara equation on a periodic domain, we show that the system is locally exactly controllable and expon...

Under the generic assumption that zero is in the resolvent set of the generator, we show that the optimal control problem for a stable well-posed linear system is equivalent to a control problem for its reciprocal system which has bounded generating operators. Consequently, the operator X that defines the optimal cost satisfies a Riccati equation with bounded operators. Previous results needed ...