Here u(x, t) represents the free surface of the liquid and the parameter γ > 0 measures the effect of rotation. (1.1) describes the propagation of internal waves of even modes in the ocean; for instance, see the work of Galkin and Stepanyants [1], Leonov [2], and Shrira [3, 4]. The parameter β determines the type of dispersion, more precisely, when β < 0, (1.1) denotes the generalized Ostrovsky...

We have recently solved the inverse scattering problem for one parameter families of vector fields, and used this result to construct the formal solution of the Cauchy problem for a class of integrable nonlinear partial differential equations connected with the commutation of multidimensional vector fields, like the heavenly equation of Plebanski, the dispersionless Kadomtsev Petviashvili (dKP)...

We consider the problem of recovering the coefficient σ (x) of the elliptic equation ▽ · (σ ▽ u) = 0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sjöstrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by so...

Let H be an infinite--dimensional Hilbert space and K(H) be the set of all compact operators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate of higher derivation and higher Jordan derivation on K(H) associated with the following cauchy-Jencen type functional equation 2f(frac{T+S}{2}+R)=f(T)+f(S)+2f(R) for all T,S,Rin K(H).

Let H be an innite dimensional Hilbert space and K(H) be the set of all compactoperators on H. We will adopt spectral theorem for compact self-adjoint operators, to investigate ofhigher derivation and higher Jordan derivation on K(H) associated with the following Cauchy-Jensentype functional equation 2f((T + S)/2+ R) = f(T ) + f(S) + 2f(R) for all T, S, R are in K(...

Solutions of the Cauchy problem for the Navier-Stokes equation, in a certain generalized sense, are not unique.

The computability of the solution operator of the Cauchy problem for the Fifth-order CamassaHolm equation is studied in this paper. Firstly, a nonlinear map KR : H → C (R;H (R)) is defined from the initial value φ to the solution u. Then we used the relevant knowledge of type-2 theory of effectivity, functional analysis and Sobolev space to prove that when s > (6 √ 10− 17) / 4, the solution ope...

The Fokker–Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker– Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leadin...

We study the relative value iteration for the ergodic control problem under a nearmonotone running cost structure for a nondegenerate diffusion controlled through its drift. This algorithm takes the form of a quasi-linear parabolic Cauchy initial value problem in Rd. We show that this Cauchy problem stabilizes or, in other words, that the solution of the quasi-linear parabolic equation converge...

where t ∈ [0, T ] for some T > 0, x ∈ T (the one-dimensional torus), and subscripts denote partial derivatives. Equation (1.1) does not admit in general classical solutions for the associated Cauchy problem, even if the initial datum is smooth. On the other hand, if f is non-linear, there exist in general infinitely many weak solutions. An admissibility condition, the so-called entropic conditi...