scholarly journals On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

2019 ◽  
Vol 150 (4) ◽  
pp. 2025-2054
Author(s):  
Piotr Kalita ◽  
Piotr Zgliczyński
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Dirichlet Boundary Conditions ◽  
Image Position ◽  
Space Interval ◽  
Bounded Trajectory

AbstractWe study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t)-u_{xx}(x,t) = f(x,t)$$ on the space interval (0, 1) with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique H1 bounded trajectory of this equation defined for all t ∈ ℝ. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

The resolvent problem for the Stokes system in exterior domains: uniqueness and non-regularity in Hölder spaces

1992 ◽  
Vol 122 (1-2) ◽  
pp. 1-10 ◽  
Author(s):  
Paul Deuring
Keyword(s):  
Boundary Conditions ◽  
Existence Result ◽  
Stokes System ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Exterior Domains ◽  
Hölder Spaces ◽  
Image Position ◽  
Holder Spaces ◽  
Resolvent Problem

SynopsisWe consider the resolvent problem for the Stokes system in exterior domains, under Dirichlet boundary conditions:where Ω is a bounded domain in ℝ3. It will be shown that in general there is no constant C > 0 such that for with , div u = 0, and for with . However, if a solution (u, π) of problem (*) exists, it is uniquely determined, provided that u(x) and ∇π(x) decay for large values of |x|. These assertions imply a non-existence result in Hölder spaces.

scholarly journals A note on the characterisation of eigencurves for certain two parameter eigenvalue problems in ordinary differential equations

1993 ◽  
Vol 35 (1) ◽  
pp. 63-67 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Weight Function ◽  
Analytic Functions ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Dirichlet Boundary Conditions ◽  
Survey Paper ◽  
Image Position ◽  
Two Parameter

We are interested in two parameter eigenvalue problems of the formsubject to Dirichlet boundary conditionsThe weight function 5 and the potential q will both be assumed to lie in L2[0,1]. The problem (1.1), (1.2) generates eigencurvesin the sense that for any fixed λ, ν(λ) is the nth eigenvalue ν, (according to oscillation indexing) of (1.1), (1.2). These curves are in fact analytic functions of λ and have been the object of considerable study in recent years. The survey paper [1] provides background in this area and itemises properties of eigencurves.

THE STOCHASTIC BURGERS EQUATION: FINITE MOMENTS AND SMOOTHNESS OF THE DENSITY

2000 ◽  
Vol 03 (03) ◽  
pp. 363-385 ◽  
Author(s):  
JORGE A. LEÓN ◽  
D. NUALART ◽  
ROGER PETTERSSON
Keyword(s):  
Boundary Conditions ◽  
Malliavin Calculus ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Gaussian White Noise ◽  
Dirichlet Boundary Conditions ◽  
Stochastic Version ◽  
Stochastic Burgers Equation ◽  
Finite Moments ◽  
Smooth Density

We apply a stochastic version of the Hopf–Cole transformation to the solution of the stochastic Burgers equation with Dirichlet boundary conditions driven by a space–time Gaussian white noise. As a consequence, we deduce that the solution of the Burgers equation has moments of all orders. Using the techniques of the Malliavin calculus we show that if the dispersion is state-independent, then the solution of the stochastic Burgers equation has a smooth density at any point (t, x), with t>0 and x ∈(0,1).

On the existence of multiple positive solutions to some superlinear systems

2012 ◽  
Vol 142 (1) ◽  
pp. 39-59 ◽  
Author(s):  
M. Chhetri ◽  
S. Raynor ◽  
S. Robinson
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Dirichlet Boundary ◽  
Upper And Lower Solutions ◽  
Elliptic Systems ◽  
Dirichlet Boundary Conditions ◽  
Multiple Positive Solutions ◽  
Convex Shape ◽  
Smooth Bounded Domain ◽  
Image Position

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the formon a smooth, bounded domain Ω⊂ℝn with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical, superlinear problems with a certain concave–convex shape to their nonlinearity.

Dynamics of a nonlinear convection-diffusion equation in multidimensional bounded domains

1995 ◽  
Vol 125 (2) ◽  
pp. 439-448 ◽  
Author(s):  
Adrian T. Hill ◽  
Endre Süli
Keyword(s):  
Diffusion Equation ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Dirichlet Boundary Conditions ◽  
Maximum Principles ◽  
Smooth Bounded Domain ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Image Position

The scalar nonlinear convection-diffusion equationis considered, for given initial data and zero Dirichlet boundary conditions, in a smooth bounded domain Ω⊂ℝn. The homogeneous viscous Burgers' equation in one dimension is well-known to possess a unique, exponentially attracting equilibrium. These properties are shown to be preserved in the generalisation considered. Furthermore, the equilibrium is shown to be bounded in the maximum norm independently of the function a. The main methods used are maximum principles, and a variational method due to Stampacchia.

scholarly journals General solutions in Chern-Simons gravity and $$ T\overline{T} $$-deformations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Eva Llabrés
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Spin Connection ◽  
Departure Point ◽  
Boundary Stress ◽  
Chern Simons

Abstract We find the most general solution to Chern-Simons AdS3 gravity in Fefferman-Graham gauge. The connections are equivalent to geometries that have a non-trivial curved boundary, characterized by a 2-dimensional vielbein and a spin connection. We define a variational principle for Dirichlet boundary conditions and find the boundary stress tensor in the Chern-Simons formalism. Using this variational principle as the departure point, we show how to treat other choices of boundary conditions in this formalism, such as, including the mixed boundary conditions corresponding to a $$ T\overline{T} $$ T T ¯ -deformation.

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