We establish the local well-posedness in H(S) with any s > 72 for a modified Camassa-Holm equation derived as the EPDiff equation with respect to the H(S) metric, and obtain the global existence of the weak solution in H(S) under some sign assumption on the initial values and prove the convergence of the corresponding finite particle approximation method.

In this paper, we study Levitin-Polyak well-posedness for vector equilibrium problems with functional constraints. Two sufficient conditions of (generalized) Levitin-Polyak well-posedness are derived for vector equilibrium problems.

In this article, we establish well-posedness and Lp-regularity of solutions to the first initial-boundary value problem for general higher order hyperbolic equations in cylinders whose base is a cusp domain.

Let Y = μ∗(X)+ε, where μ∗ is unknown and E[ε|X] 6= 0 with positive probability but there exist instrumental variables W such that E[ε|W ] = 0 w.p.1. It is well known that such nonparametric regression models are generally “ill-posed” in the sense that the map from the data to μ∗ is not continuous. In this paper, we derive the efficiency bounds for estimating certain linear functionals of μ∗ wit...

We prove the invariance of the Gibbs measure for the periodic SchrödingerBenjamin-Ono system (when the coupling parameter |γ| 6= 0, 1) by establishing a new local well-posedness in a modified Sobolev space and constructing the Gibbs measure (which is in the sub-L setting for the Benjamin-Ono part.) We also show the ill-posedness result in H(T)×H 1 2 (T) for s < 1 2 when |γ| 6= 0, 1 and for any ...

We propose a systemof partial differential equations with a single constant delay τ > 0 describing the behavior of a one-dimensional thermoelastic solid occupying a bounded interval of R. For an initial-boundary value problem associated with this system, we prove a well-posedness result in a certain topology under appropriate regularity conditions on the data. Further, we show the solution of o...

The system is called a Benjamin-Ono-Boussinesq system because it can be reduced to a pair of equations whose linearization uncouples to a pair of linear Benjamin-Ono equations. Equations of type (1.1) are a class of essential model equations appearing in physics and fluid mechanics. To describe two-dimensional irrotational flows of an inviscid liquid in a uniform rectangular channel, Boussinesq...

We study the initial value problem for the quasi-geostrophic type equations ∂θ ∂t + u · ∇θ + (−∆)θ = 0, on R × (0,∞), θ(x, 0) = θ0(x), x ∈ R n , where λ(0 ≤ λ ≤ 1) is a fixed parameter and u = (uj) is divergence free and determined from θ through the Riesz transform uj = ±Rπ(j)θ, with π(j) a permutation of 1, 2, · · · , n. The initial data θ0 is taken in the Sobolev space L̇r,p with negative ind...

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the...