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Register a new userInertial manifolds for 1D reaction-diffusion-advection systems. Part I: Dirichlet and Neumann boundary conditions
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This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of the dependent variable. The case of periodic boundary conditions where the situation is principally different and the inertial manifold may not exist is considered in the second part of our study.
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This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in cite{KZI} and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed by periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for this class of dissipative systems strongly depend on the choice of boundary conditions.
In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion $partial_t u = text{div}(k(x) abla G(u))$, $u|_{t=0}=u_0$ with Neumann boundary conditions $k(x) abla G(u)cdot u = 0$. Here $xin Bsubset mathbb{R}^d$, a bounded open set with locally Lipchitz boundary, and with $ u$ as the unit outer normal. The function $G$ is Lipschitz continuous and nondecreasing, while $k(x)$ is diagonal matrix. We show that any two weak entropy solutions $u$ and $v$ satisfy $Vert{u(t)-v(t)}Vert_{L^1(B)}le Vert{u|_{t=0}-v|_{t=0}}Vert_{L^1(B)}e^{Ct}$, for almost every $tge 0$, and a constant $C=C(k,G,B)$. If we restrict to the case when the entries $k_i$ of $k$ depend only on the corresponding component, $k_i=k_i(x_i)$, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.
We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is, instead of using local gradients of the boundary data that are not available a priori, we incorporate nonlocal gradient operators into the formulation to generalize the finite differences-based methods which are pervasive in literature; our particular choice of the nonlocal gradient operators is based on the interplay between a constant kernel function and the geometry of nonlocal interaction neighborhoods. Such an approach can be potentially useful to address similar issues in peridynamics, smoothed particle hydrodynamics and other nonlocal models. We first show the well-posedness of the newly formulated nonlocal problems and then analyze their asymptotic convergence to the local limit as the nonlocality parameter shrinks to zero. We justify the second order localization rate, which is the optimal order attainable in the absence of physical boundaries.
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.
We prove the existence of an Inertial Manifold for 3D complex Ginzburg-Landau equation with periodic boundary conditions as well as for more general cross-diffusion system assuming that the dispersive exponent is not vanishing. The result is obtained under the assumption that the parameters of the equation is chosen in such a way that the finite-time blow up of smooth solutions does not take place. For the proof of this result we utilize the recently suggested method of spatio-temporal averaging.
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