scholarly journals Weak well-posedness of multidimensional stable driven SDEs in the critical case

2020 ◽  
Vol 20 (06) ◽  
pp. 2040004
Author(s):  
Paul-Éric Chaudru de Raynal ◽  
Stéphane Menozzi ◽  
Enrico Priola
Keyword(s):  
Critical Case ◽  
Additive Noise ◽  
Well Posedness ◽  
One Dimensional ◽  
Drift Term ◽  
Cauchy Processes ◽  
Stable Index

We establish weak well-posedness for critical symmetric stable driven SDEs in [Formula: see text] with additive noise [Formula: see text], [Formula: see text]. Namely, we study the case where the stable index of the driving process [Formula: see text] is [Formula: see text] which exactly corresponds to the order of the drift term having the coefficient [Formula: see text] which is continuous and bounded. In particular, we cover the cylindrical case when [Formula: see text] and [Formula: see text] are independent one-dimensional Cauchy processes. Our approach relies on [Formula: see text]-estimates for stable operators and uses perturbative arguments.

Error feedback regulation for 1D anti-stable wave equation

2021 ◽  
pp. 014233122110592
Author(s):  
Zhiyuan Li ◽  
Feng-Fei Jin
Keyword(s):  
Wave Equation ◽  
Tracking Error ◽  
Feedback Regulation ◽  
Original System ◽  
Well Posedness ◽  
Error Feedback ◽  
One Dimensional ◽  
Internal Loop ◽  
Finite Dimensional ◽  
Uniformly Bounded

This paper is concerned with the boundary error feedback regulation for a one-dimensional anti-stable wave equation with distributed disturbance generated by a finite-dimensional exogenous system. Transport equation and regulator equation are introduced first to deal with the anti-damping on boundary and the distributed disturbance of the original system. Then, the tracking error and its derivative are measured to design an observer for both exosystem and auxiliary partial differential equation (PDE) system to recover the state. After proving the well-posedness of the regulator equations, we propose an observer-based controller to regulate the tracking error to zero exponentially and keep the states of all the internal loop uniformly bounded. Finally, some numerical simulations are presented to validate the effectiveness of the proposed controller.

scholarly journals The Periodic Boundary Value Problem for the Weakly Dissipativeμ-Hunter-Saxton Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Zhengyong Ouyang ◽  
Xiangdong Wang ◽  
Haiwu Rong
Keyword(s):  
Boundary Value Problem ◽  
Besov Space ◽  
Periodic Boundary ◽  
Critical Case ◽  
Boundary Value ◽  
Periodic Boundary Value Problem ◽  
Well Posedness

We study the periodic boundary value problem for the weakly dissipativeμ-Hunter-Saxton equation. We establish the local well-posedness in Besov spaceB2,13/2, which extends the previous regularity range to the critical case.

The non-autonomous wave equation with general Wentzell boundary conditions

2005 ◽  
Vol 135 (2) ◽  
pp. 317-329 ◽  
Author(s):  
Angelo Favini ◽  
Ciprian G. Gal ◽  
Gisèle Ruiz Goldstein ◽  
Jerome A. Goldstein ◽  
Silvia Romanelli
Keyword(s):  
Hilbert Space ◽  
Wave Equation ◽  
Boundary Conditions ◽  
Abstract Cauchy Problem ◽  
Regularity Conditions ◽  
Well Posedness ◽  
One Dimensional ◽  
Wentzell Boundary Conditions ◽  
Image Position ◽  
Dimensional Wave

We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation utt = A(t)u with general Wentzell boundary conditions Here A(t)u := (a(x, t)ux)x, a(x, t) ≥ ε > 0 in [0, 1] × [0, + ∞) and βj(t) > 0, γj(t) ≥ 0, (γ0(t), γ1(t)) ≠ (0,0). Under suitable regularity conditions on a, βj, γj we prove the well-posedness in a suitable (energy) Hilbert space

scholarly journals Well-posedness of the linear heat equation with a second order memory term

2020 ◽  
Vol 34 ◽  
pp. 03011
Author(s):  
Constantin Niţă ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Unique Solution ◽  
Source Term ◽  
Memory Term ◽  
Well Posedness ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Linear Heat ◽  
The One

In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.

scholarly journals Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kiyeon Lee
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Critical Case ◽  
Nonlinear Term ◽  
Well Posedness ◽  
Low Regularity ◽  
Bilinear Estimates ◽  
Dirac Equations ◽  
The Cauchy Problem

<p style='text-indent:20px;'>In this paper, we consider the Cauchy problem of <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimension Hartree type Dirac equation with nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ c\in \mathbb R\setminus\{0\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 &lt; \gamma &lt; d $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M5">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>). Our aim is to show the local well-posedness in <inline-formula><tex-math id="M6">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ s &gt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula> with mass-supercritical cases(<inline-formula><tex-math id="M8">\begin{document}$ 1 &lt; \gamma&lt;d $\end{document}</tex-math></inline-formula>) and mass-critical case(<inline-formula><tex-math id="M9">\begin{document}$ {\gamma} = 1 $\end{document}</tex-math></inline-formula>) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be <inline-formula><tex-math id="M10">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> at the origin for <inline-formula><tex-math id="M11">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M12">\begin{document}$ s &lt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals On the global well-posedness of the one-dimensional Schrödinger map flow

2009 ◽  
Vol 2 (2) ◽  
pp. 187-209 ◽  
Author(s):  
Igor Rodnianski ◽  
Yanir Rubinstein ◽  
Gigliola Staffilani
Keyword(s):  
Well Posedness ◽  
One Dimensional ◽  
The One ◽  
Schrödinger Map
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