scholarly journals On the Sharp Gårding Inequality for Operators with Polynomially Bounded and Gevrey Regular Symbols

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1938
Author(s):  
Alexandre Arias Junior ◽  
Marco Cappiello
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Expansion ◽  
Evolution Equations ◽  
Gevrey Estimates ◽  
The Cauchy Problem ◽  
Precise Expression

In this paper, we analyze the Friedrichs part of an operator with polynomially bounded symbol. Namely, we derive a precise expression of its asymptotic expansion. In the case of symbols satisfying Gevrey estimates, we also estimate precisely the regularity of the terms in the asymptotic expansion. These results allow new and refined applications of the sharp Gårding inequality in the study of the Cauchy problem for p-evolution equations.

scholarly journals Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions

2020 ◽  
Vol 23 (6) ◽  
pp. 1663-1677
Author(s):  
Michael Ruzhansky ◽  
Berikbol T. Torebek
Keyword(s):  
Cauchy Problem ◽  
Exponential Function ◽  
Evolution Equations ◽  
Oscillatory Integrals ◽  
Schrödinger Equations ◽  
Fractional Evolution Equations ◽  
Type Function ◽  
Fractional Schrödinger Equations ◽  
The Cauchy Problem ◽  
Klein Gordon

Abstract The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E α, β (i λ ϕ(x)), x ∈ ℝ N and E α, β (i α λ ϕ(x)), x ∈ ℝ N for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

scholarly journals Double asymptotic expansion of the resolving operator of the cauchy problem for the line-arized system of gas dynamics

2019 ◽  
Vol 484 (2) ◽  
pp. 134-137
Author(s):  
A. I. Allilueva ◽  
A. I. Shafarevich
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Expansion ◽  
Gas Dynamics ◽  
Acoustic Modes ◽  
Small Viscosity ◽  
The Cauchy Problem ◽  
Linearized System

We obtain double asymptotic expansion (with respect to smoothness and small viscosity) of the resolving operator of the Cauchy problem for the linearized system of gas dynamics. We derive estimates for the summands and for the residual in the Sobolev scale.We describe explicitly hydrodynamic and acoustic modes.

A note on optimal L 2 -estimates of solutions to some strongly damped σ-evolution equations

2020 ◽  
Vol 121 (1) ◽  
pp. 59-74
Author(s):  
Ryo Ikehata
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Evolution Equations ◽  
Estimates Of Solutions ◽  
The Cauchy Problem ◽  
Strongly Damped

We consider the Cauchy problem in R n for the so-called σ-evolution equations with damping terms. We derive asymptotic profiles of solutions with weighted L 1 , 1 ( R n ) initial data, and investigates the optimality of estimates of solutions in L 2 -sense. The obtained results will generalize and compensate those already known in (J. Math. Anal. Appl. 478 (2019) 476–498, J. Diff. Eqns 257 (2014) 2159–2177, Diff. Int. Eqns 30 (2017), 505–520).

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