The aim of this short note is to reconsider and to extend a former result of K. Mochizuki [Moc76], [MN96] on the existence of the scattering operator for wave equations with small dissipative terms. Contrary to the approach used by Mochizuki we construct the wave operator explicitly in terms of the parametrix construction obtained by a (simplified) diagonalization procedure, cf. [Yag97]. The me...

A subsequence principle is obtained, characterizing Banach spacescontaining co' in the spirit of the author's 1974 characterization of Banachspaces containing II . Defiaition. A sequence(bj )in a Banach space is called strongly summing (s.s.)if (bj ) is a weak-Cauchy basic sequence so that whenever scalars (cj ) satisfysUPn II E~=I cijll < 00, thenECj converges.A...

|x−y|≤ǫ |x−y| |V (y)|dy = 0) compared to the classical Cauchy-Lipshitz condition for vector fields. On the other hand, tremendous progress have been done in the last 25 years concerning the theory of ODEs using PDE’s methods: extension of the Cauchy-Lipshitz condition to Sobolev ones (DiPerna-Lions 1989) and BV vector fields (Bouchut for the Hamiltonian case 2001 and Ambrosio for the general ca...

In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. Furthermore, we prove that all even order generalized Cauchy tensorsw...

We review different properties related to the Cauchy problem for the (nonlinear) Schrödinger equation with a smooth potential. For energy-subcritical nonlinearities and at most quadratic potentials, we investigate the necessary decay in space in order for the Cauchy problem to be locally (and globally) well-posed. The characterization of the minimal decay is different in the case of super-quadr...

We propose a discretized Tikhonov regularization for a Cauchy problem for an elliptic equation by a reproducing kernel Hilbert space. We prove the convergence of discretized regularized solutions to an exact solution. Our numerical results demonstrate that our method can stably reconstruct solutions to the Cauchy problems even in severe cases of geometric configurations.

We prove two new reverse Cauchy–Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C *-algebra. We apply our results to get some norm and integral inequalities. As a consequence, we improve a celebrated reverse Cauchy–Schwarz inequality due to G. Pólya and G. Szegö.

In this paper we study the concepts of I-Cauchy and I∗-Cauchy for double sequences in a linear metric space. Also, we give the relation between I-convergence and I∗-convergence of double sequences of functions defined between linear metric spaces.

The paper studies some ill-posed boundary value problems on semi-plane for parabolic equations with homogenuous Cauchy condition at initial time and with the second order Cauchy condition on the boundary of the semi-plane. A class of inputs that allows some regularity is suggested and described explicitly in frequency domain. This class is everywhere dense in the space of square integrable func...

We study global well-posedness of strong solutions to the Cauchy problem nonhomogeneous heat conducting Navier–Stokes equations with vacuum on whole space $${\mathbb {R}}^3$$ . derive existence and uniqueness provided that $$\Vert \rho _0\Vert _{L^\infty }\Vert \sqrt{\rho _0}{\mathbf {u}}_0\Vert _{L^2}^2\Vert \nabla {\mathbf _{L^2}^2$$ is suitably small, smallness depending only viscosity coeff...