WAVE EQUATION WITH SECOND-ORDER NON-STANDARD DYNAMICAL BOUNDARY CONDITIONS

2008 ◽  
Vol 18 (12) ◽  
pp. 2019-2054 ◽  
Author(s):  
JUAN LUIS VAZQUEZ ◽  
ENZO VITILLARO
Keyword(s):  
Space Variable ◽  
Regularity Theory ◽  
Laplacian Operator ◽  
Well Posedness ◽  
Open Domain ◽  
One Dimensional ◽  
Ill Posed ◽  
Dynamical Boundary Conditions ◽  
The One ◽  
Well Posed

The paper deals with the well-posedness of the problem [Formula: see text] where u = u(t, x), t ∈ ℝ, x ∈ Ω, Δ = Δx denotes the Laplacian operator with respect to the space variable, Ω is a bounded regular (C∞) open domain of ℝN (N ≥ 1), Γ = ∂Ω, ν is the outward normal to Ω, k is a constant. We prove that it is ill-posed if N ≥ 2, while it is well-posed when N = 1. In the one-dimensional case, we give a complete existence, uniqueness and regularity theory. We also give some existence result for regular initial data when N ≥ 2 and Ω is a ball.

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 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

scholarly journals A note on inverse problem for strongly damped wave equation with Gaussian white noise

2018 ◽  
Vol 20 ◽  
pp. 02003
Author(s):  
Chu Duc Khanh ◽  
Nguyen Hoang Luc ◽  
Van Phan ◽  
Nguyen Huy Tuan
Keyword(s):  
White Noise ◽  
A Priori ◽  
Gaussian White Noise ◽  
True Solution ◽  
One Dimensional ◽  
Noise Data ◽  
Ill Posed ◽  
The One ◽  
First Time ◽  
Strongly Damped

In this paper, we study for the first time the inverse initial problem for the one-dimensional strongly damped wave with Gaussian white noise data. Under some a priori assumptions on the true solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Error estimates are given in both the L2– and Hp–norms.

scholarly journals On non-resistive limit of 1D MHD equations with no vacuum at infinity

2021 ◽  
Vol 11 (1) ◽  
pp. 702-725
Author(s):  
Zilai Li ◽  
Huaqiao Wang ◽  
Yulin Ye
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Strong Solutions ◽  
Mhd Equations ◽  
Well Posedness ◽  
Large Initial Data ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Global Strong Solutions ◽  
The One

Abstract In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

scholarly journals Ill posedness in modelling two-dimensional morphodynamic problems: effects of bed slope and secondary flow

2019 ◽  
Vol 868 ◽  
pp. 461-500 ◽  
Author(s):  
Víctor Chavarrías ◽  
Ralph Schielen ◽  
Willem Ottevanger ◽  
Astrid Blom
Keyword(s):  
Sediment Transport ◽  
Numerical Simulations ◽  
Secondary Flow ◽  
Vertical Mixing ◽  
Well Posedness ◽  
Two Dimensional ◽  
Transport Direction ◽  
Flow Curvature ◽  
Ill Posed ◽  
Well Posed

A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.

scholarly journals The persistence of logconcavity for positive solutions of the one dimensional heat equation

1990 ◽  
Vol 48 (2) ◽  
pp. 246-263 ◽  
Author(s):  
G. Keady
Keyword(s):  
Maximum Principle ◽  
Heat Equation ◽  
Positive Solutions ◽  
Space Variable ◽  
Time Variable ◽  
One Dimensional ◽  
The Maximum Principle ◽  
Proof Techniques ◽  
The One

AbstractConsider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

scholarly journals Thermoelasticity of Moore–Gibson–Thompson type with history dependence in the temperature

2020 ◽  
Vol 120 (1-2) ◽  
pp. 1-21 ◽  
Author(s):  
Monica Conti ◽  
Vittorino Pata ◽  
Ramon Quintanilla
Keyword(s):  
Heat Conduction ◽  
Differential System ◽  
Exponential Decay ◽  
Relaxation Parameter ◽  
Dimensional Case ◽  
Well Posedness ◽  
History Dependence ◽  
One Dimensional ◽  
Thermoelastic Model ◽  
The One

In this paper, we consider a thermoelastic model where heat conduction is described by the history dependent version of the Moore–Gibson–Thompson equation, arising via the introduction of a relaxation parameter in the Green-Naghdi type III theory. The well-posedness of the resulting integro-differential system is discussed. In the one-dimensional case, the exponential decay of the energy is proved.

Estimates for the controls of the wave equation with a potential

2018 ◽  
Vol 24 (1) ◽  
pp. 289-309 ◽  
Author(s):  
Sorin Micu ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Space Variable ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
One Dimensional ◽  
Boundary Controls ◽  
The One ◽  
Variable Potential ◽  
Uniformly Bounded

This article studies the L2-norm of the boundary controls for the one dimensional linear wave equation with a space variable potential a = a(x). It is known these controls depend on a and their norms may increase exponentially with ||a||L∞. Our aim is to make a deeper study of this dependence in correlation with the properties of the initial data. The main result of the paper shows that the minimal L2−norm controls are uniformly bounded with respect to the potential a, if the initial data have only sufficiently high eigenmodes.

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