This paper is devoted to giving definitions of Besov spaces on an arbitrary open set Omega of R-n via the spectral theorem for the Schrodinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spaces on Omega. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schrodinger operators. (C) 2019 Elsevier Masson SAS. All rights reserved.

In the whole space R-d, linear estimates for heat semigroup in Besov spaces are well established, which are estimates of L-p - L-q type, with maximal regularity, etc. This paper is concerned with such estimates for the semigroup generated by the Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of R-d.

Let H-v be a self-adjoint extension of the Schrodinger operator -Delta + V(x) with the Dirichlet boundary condition on an arbitrary open set Omega of R-d, where d >= 1 and the negative part of potential V belongs to the Kato class on Omega. The purpose of this paper is to prove L-P-L-q estimates and gradient estimates for an operator phi(H-v), where (p is an arbitrary rapidly decreasing function on R, and phi(H-v) is defined via the spectral theorem.

We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation,partial derivative(t)u + (-Delta)(1/2)u = vertical bar del u vertical bar(p), x is an element of R-N, t < 0, u(x,0) = u(0)(x), x is an element of R-N,where p < 1 and u(0) is an element of Beta R-1(r,1()N) boolean AND B-infinity,1(1)(R-N) with r is an element of [1,infinity]. We show that for sufficiently small u(0) is an element of Beta(1)(infinity,1)(R-N), there exists a global-in-time mild solution. Furthermore, we prove that the solution behaves asymptotically like suitable multiplies of the Poisson kernel. (C) 2016 Elsevier Masson SAS. All rights reserved.

We consider the initial value problem of the 3D incompressible Boussinesq equations for rotating stratified fluids. We establish the dispersive and the Strichartz estimates for the linear propagator associated with both the rotation and the stable stratification. As an application, we give a simple proof of a unique existence of global in time solutions to our system using just a contraction mapping principle. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The existence time of local solutions of semilinear Schrodinger equations in Sobolev spaces is considered based on the method of frequency decomposition. The semilinear terms are power type or exponential type, which are critical in terms of the scaling or the Trudinger-Moser inequality. The existence time is estimated by the low and high frequency parts of data. (C) 2016 Elsevier Ltd. All rights reserved.

The existence time of local solutions of semilinear Schrödinger equations in Sobolev spaces is considered based on the method of frequency decomposition. The semilinear terms are power type or exponential type, which are critical in terms of the scaling or the Trudinger–Moser inequality. The existence time is estimated by the low and high frequency parts of data. © 2016 Elsevier Ltd

This paper concerns the ill-posedness issue for a system of quadratic nonlinear Schrodinger equations in two dimensions. From previous studies for the large time behavior of the solution to the system, one may expect that the critical regularity to show the well-posedness depends on the dispersion coefficients. To prove the ill-posedness, we show the failure of the uniform continuity of the data-solution map for the mass resonance case and show the norm inflation for other cases. (C) 2016 The Authors. Published by Elsevier Inc.

We are concerned with the ill-posedness issue for the nonlinear Schrodinger equation with the quadratic nonlinearity vertical bar u vertical bar(2) and prove the norm inflation in the dimensions 1 <= n <= 3. This is the extension of the ill-posed result by Kishimoto-Tsugawa [12] in one dimension and also the remaining case of Iwabuchi-Ogawa [7].

Abstract We consider the Cauchy problem for the critical Burgers equation. The existence and the uniqueness of global solutions for small initial data are studied in the Besov space B0∞,1(&Rdbl;n) and it is shown that the global solutions are bounded in time. We also study the large time behavior of the solutions with the initial data u0∈L1(&Rdbl;n)∩B0∞,1(&Rdbl;n) to show that the solution behaves like the Poisson kernel. © 2014 Elsevier Masson SAS.