In this work, we compare a parabolic equation with an elliptic equation both of which are used in modeling temperature profile of a powerlaw polymer flow in a semi-infinite straight pipe with circular cross section. We show that both models are well-posed and we derive exponential rates of convergence of the two solutions to the same steady state solution away from the entrance. We also show es...

Abstract. In this paper we investigate an optimal control problem in the space of measures on R. The problem is motivated by a stochastic interacting particle model which gives the 2-D Navier-Stokes equations in their vorticity formulation as mean-field equation. We prove that the associated Hamilton-Jacobi-Bellman equation, in the space of probability measures, is well-posed in an appropriatel...

Considering the Cauchy problem for the modified Korteweg-de Vries-Burgers equation ut + uxxx + ǫ|∂x| u = 2(u)x, u(0) = φ, where 0 < ǫ, α ≤ 1 and u is a real-valued function, we show that it is uniformly globally well-posed in Hs (s ≥ 1) for all ǫ ∈ (0, 1]. Moreover, we prove that for any s ≥ 1 and T > 0, its solution converges in C([0, T ]; Hs) to that of the MKdV equation if ǫ tends to 0.

In this paper, we study the Ostrovsky, Stepanyams and Tsimring equation. We show that the associated initial value problem is locally well-posed in Sobolev spaces H (R) for s > −3/2. We also prove that our result is sharp in the sense that the flow map of this equation fails to be C in H(R) for s < −3/2. AMS subject classifications: 35A07, 35Q53, 35Q35

Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundaryvalue problem for a general Schrödinger-type equation posed on a two space dimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain and we approximate its solut...

The heat equation posed on the half-line may be used as a simple mathematical model describing the operation of an amperometric ion sensor. These sensors represent the next generation of sensors that are in routine use today. Such sensors may be used to measure ion concentrations in the laboratory, for clinical analysis, environmental monitoring, process and quality control, biomedical analysis...

We consider an aggregation equation in Rn, n ≥ 2, with fractional dissipation, namely, ut + ∇ · (u∇K ∗ u) = −ν(−∆)γ/2u , where 0 ≤ γ ≤ 2 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e−|x|. We prove that for 0 ≤ γ < 1 the solutions develop blow-up in finite for a general class of initial data. In contrast we prove that for 1 < γ ≤ 2 the equati...

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. A modified Tikhonov regularization method is presented to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. Numerical examples show that the proposed method is effective and stable....

We address in this work some theoretical and numerical issues on optimal design problems concerning evolution equation and more specifically the wave equation posed in 1-D and 2-D domain. We compute the gradient of the cost function and then use the level set approach to optimize the location of actuators in order to stabilize or control exactly a given system Copyright c ©2005 IFAC.

The evolution of an elastic-plastic material is modeled as an initial boundary value problem consisting of the dynamic momentum equation coupled with a constitutive law for which the hysteretic dependence between stress and strain is described by a system of variational inequalities. This system is posed as an evolution equation in Hilbert space for which is proved the existence and uniqueness ...