In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-intime existence of classical solution to the initial-boundary value problem by the method of energy estimates. keywords: Euler equation; initial-boundary value problem; well-posedness. MSC(2000): 35A05; 35L45.

Considering the Cauchy problem for the modified finite-depthfluid equation ∂tu− Gδ(∂ 2 xu)∓ u 2ux = 0, u(0) = u0, where Gδf = −iF [coth(2πδξ)− 1 2πδξ ]Ff , δ&1, and u is a real-valued function, we show that it is uniformly globally well-posed if u0 ∈ Hs (s ≥ 1/2) with ‖u0‖L2 sufficiently small for all δ&1. Our result is sharp in the sense that the solution map fails to be C in Hs(s < 1/2). More...

We study a class of equilibrium problems which is identified with a complete metric space of functions. For most elements of this space of functions in the sense of Baire category, we establish that the corresponding equilibrium problem possesses a unique solution and is well-posed.

We consider the problem of hedging the loss of a given portfolio of derivatives using a set of more liquid derivative instruments. We illustrate why the typical mathematical formulation for this hedging problem is ill-posed. We propose to determine a hedging portfolio by minimizing a proportional cost subject to an upper bound on the hedge risk; this bound is typically slightly larger than the ...

Reconstruction of a signal from its spectral phase or magnitude is in general an ill-posed problem. Various conditions restricting the class of signals under consideration have been shown to be su cient to regularize the problem so that a unique (or essentially unique) signal corresponds to any given spectral magnitude or spectral phase function. This paper shows that a nite discretetime signal...

Theorems on the Fredholm alternative and well-posedness of the Darboux problem ∂2u(t, x) ∂t ∂x = `(u)(t, x) + q(t, x), u(t, x0) = φ(t) for t ∈ [a, b], u(t0, x) = ψ(x) for x ∈ [c, d] are established, where ` : C(D;R) → L(D;R) is a linear bounded operator, q ∈ L(D;R), t0 ∈ [a, b], x0 ∈ [c, d], φ : [a, b]→ R, ψ : [c, d]→ R are absolutely continuous functions, and D = [a, b] × [c, d]. New sufficien...

We analyze a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic PDE system. The well-posedness of the diffuse domain approximation is shown using weighted Sobolev spaces and we prove that the solution to the diffuse domain approximation converges weakly to the solution of the coupled bulk-surface elliptic system as the approxima...