In this paper, the global solvability of the initial boundary value problem and the periodic problem are discussed for a doublediffusive convection system under the homogeneous Neumann boundary condition in a bounded domain. This system is coupled with the so-called Brinkman-Forchheimer equations, which is similar to the Stokes equations and contains some convection terms similar to that in the Navier-Stokes equations. However, in contrast to the Navier-Stokes equations, it is shown that the global solvability in L 2 -spaces holds true for the 3-dimensional problems.
Communicated by M. EfendievIn this paper, we consider the existence of global attractor and exponential attractor for some dynamical system generated by nonlinear parabolic equations in bounded domains with the dimension N Ä 4 which describe double-diffusive convection phenomena in a porous medium. We deal with both of homogeneous Dirichlet and Neumann boundary condition cases. Especially, when Neumann condition is imposed, we need some assumptions and restrictions for the external forces and the average of initial data, since the mass conservation law holds.
Let V be a finite set, E ⊂ 2 V be a set of hyperedges, and w ∶ E → (0, ∞) be an edge weight. On the (wighted) hypergraph G = (V, E, w), we can define a multivalued nonlinear operator L G,p (p ∈ [1, ∞)) as the subdifferential of a convex function on R V , which is called "hypergraph p-Laplacian." In this article, we first introduce an inequality for this operator L G,p , which resembles the Poincaré-Wirtinger inequality in PDEs. Next, we consider an ordinary differential equation on R V governed by L G,p , which is referred as "heat" equation on the graph and used to study the geometric structure of the hypergraph in recent researches. With the aid of the Poincaré-Wirtinger type inequality, we can discuss the existence and the large time behavior of solutions to the ODE by procedures similar to those for the standard heat equation in PDEs with the zero Neumann boundary condition.
We are concerned with the time-periodic problem of some doubly nonlinear equations governed by differentials of two convex functionals over uniformly convex Banach spaces. Akagi-Stefanelli (2011) [4] considered Cauchy problem of the same equation via the so-called WED functional approach. Main purpose of this paper is to show the existence of the time-periodic solution under the same growth conditions on functionals and differentials as those imposed in [4]. Because of the difference of nature between Cauchy problem and the periodic problem, we can not apply the WED functional approach directly, so we here adopt standard compactness methods with suitable approximation procedures.
<p style='text-indent:20px;'>In this paper, we consider a doubly nonlinear parabolic equation <inline-formula><tex-math id="M2">\begin{document}$ \partial _t \beta (u) - \nabla \cdot \alpha (x , \nabla u) \ni f $\end{document}</tex-math></inline-formula> with the homogeneous Dirichlet boundary condition in a bounded domain, where <inline-formula><tex-math id="M3">\begin{document}$ \beta : \mathbb{R} \to 2 ^{ \mathbb{R} } $\end{document}</tex-math></inline-formula> is a maximal monotone graph satisfying <inline-formula><tex-math id="M4">\begin{document}$ 0 \in \beta (0) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \nabla \cdot \alpha (x , \nabla u ) $\end{document}</tex-math></inline-formula> stands for a generalized <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian. Existence of solution to the initial boundary value problem of this equation has been studied in an enormous number of papers for the case where single-valuedness, coerciveness, or some growth condition is imposed on <inline-formula><tex-math id="M7">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>. However, there are a few results for the case where such assumptions are removed and it is difficult to construct an abstract theory which covers the case for <inline-formula><tex-math id="M8">\begin{document}$ 1 < p < 2 $\end{document}</tex-math></inline-formula>. Main purpose of this paper is to show the solvability of the initial boundary value problem for any <inline-formula><tex-math id="M9">\begin{document}$ p \in (1, \infty ) $\end{document}</tex-math></inline-formula> without any conditions for <inline-formula><tex-math id="M10">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> except <inline-formula><tex-math id="M11">\begin{document}$ 0 \in \beta (0) $\end{document}</tex-math></inline-formula>. We also discuss the uniqueness of solution by using properties of entropy solution.</p>
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