General Unbounded Domains Research Articles

Asymptotic analysis of the 2D and 3D convective Brinkman–Forchheimer equations in unbounded domains

In this work, we carry out the asymptotic analysis of two- and three-dimensional convective Brinkman–Forchheimer (CBF) equations, which characterize the motion of incompressible fluid flows in a saturated porous medium. We establish the existence of a global attractor in both bounded (using compact embedding) and unbounded Poincaré domains (using asymptotic compactness property). In Poincaré domains, the estimates for Hausdorff as well as fractal dimensions of the global attractors are also obtained. We then show an upper semicontinuity of global attractors with respect to domains for CBF equations. We consider an expanding sequence of simply connected, bounded and smooth subdomains [Formula: see text] of the Poincaré domain [Formula: see text] such that [Formula: see text] as [Formula: see text]. If [Formula: see text] and [Formula: see text] are the global attractors of CBF equations corresponding to [Formula: see text] and [Formula: see text], respectively, then we show that for large enough [Formula: see text], the global attractor [Formula: see text] enters into any neighborhood [Formula: see text] of [Formula: see text]. The presence of the Darcy term in CBF equations helps us to obtain the above mentioned results in general unbounded domains also. Finally, we discuss the quasi-stability property of the semigroup associated with CBF equations in bounded domains and establish the existence of finite fractal dimensional global as well as exponential attractors.

The Helmholtz decomposition of a $BMO$ type vector field in general unbounded domains

The Helmholtz decomposition of a $BMO$ type vector field in general unbounded domains

Attractors for semigroups with multi-dimensional time and PDEs in unbounded domains

We develop the attractors theory for the semigroups with multidimensional time belonging to some closed cone in an Euclidean space and apply the obtained general results to partial differential equations (PDEs) in unbounded domains. The main attention is payed to elliptic boundary problems in general unbounded domains. In contrast to the previous works in this direction our theory does not require the underlying domain to be cylindrical or cone-like or to be shift semiinvariant with respect to some direction. In particular, the theory is applicable to the exterior domains.

Carleson perturbations for the regularity problem

We prove that the solvability of the regularity problem in L^q(\partial \Omega) is stable under Carleson perturbations. If the perturbation is small, then the solvability is preserved in the same L^q , and if the perturbation is large, the regularity problem is solvable in L^{r} for some other r\in (1,\infty) . We extend an earlier result from Kenig and Pipher to very general unbounded domains, possibly with lower dimensional boundaries as in the theory developed by Guy David and the last two authors. To be precise, we only need the domain to have non-tangential access to its Ahlfors regular boundary, together with a notion of gradient on the boundary.

Liouville property of fractional Lane-Emden equation in general unbounded domain

Abstract Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation ( − Δ ) α u = u p i n Ω , u = 0 i n R N ∖ Ω , $$\begin{array}{} \displaystyle (-{\it\Delta})^\alpha u = u^p\quad {\rm in}\quad {\it\Omega},\qquad u = 0\quad {\rm in}\quad \mathbb{R}^N\setminus {\it\Omega}, \end{array}$$ where α ∈ (0, 1), N ≥ 1, p > 0 and Ω ⊂ ℝ N–1 × [0, +∞) is an unbounded domain satisfying that Ωt := {x′ ∈ ℝ N–1 : (x′, t) ∈ Ω} with t ≥ 0 has increasing monotonicity, that is, Ωt ⊂ Ω t′ for t′ ≥ t. The shape of Ω ∞ := lim t→∞ Ωt in ℝ N–1 plays an important role to obtain the nonexistence of positive solutions for the fractional Lane-Emden equation.

Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications

In this paper, we establish various maximum principles and develop the method of moving planes and the sliding method (on general unbounded domains) for equations involving the uniformly elliptic nonlocal Bellman operator. As a consequence, we derive multiple applications of these maximum principles and the moving planes method. For instance, we prove symmetry, monotonicity and uniqueness results and asymptotic properties for solutions to various equations involving the uniformly elliptic nonlocal Bellman operator in bounded domains, unbounded domains, epigraph or $\mathbb{R}^{n}$. In particular, the uniformly elliptic nonlocal Monge-Amp\`{e}re operator introduced by Caffarelli and Charro in \cite{CC} is a typical example of the uniformly elliptic nonlocal Bellman operator.

Algebraic decay of weak solutions to 3D Navier-Stokes equations in general unbounded domains

In this paper, we study the decay properties of a weak solution to the nonstationary Navier-Stokes equations by means of the spectral decomposition method of fractional powers of the Stokes operator, in an arbitrary unbounded domain. The explicit L2-decay rates are given as t⟶∞, which improves greatly the decay results in [8,9].

Stability analysis for semilinear parabolic problems in general unbounded domains

We introduce several notions of generalised principal eigenvalue for a linear elliptic operator on a general unbounded domain, under boundary condition of the oblique derivative type. We employ these notions in the stability analysis of semilinear problems. Some of the properties we derive are new even in the Dirichlet or in the whole space cases. As an application, we show the validity of the hair-trigger effect for the Fisher-KPP equation on general, uniformly smooth domains.

Maximal regularity of the Stokes system with Navier boundary condition in general unbounded domains

Consider the instationary Stokes system in general unbounded domains $\Omega \subset \mathbb{R}^n$, $n \geq 2$, with boundary of uniform class $C^3$, and Navier slip or Robin boundary condition. The main result of this article is the maximal regularity of the Stokes operator in function spaces of the type $\tilde{L}^q$ defined as $L^q \cap L^2$ when $q \geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain.

Extremal solutions of logistic-type equations in exterior domain in [formula omitted

Let Ω=R2∖B(0,1)¯ be the exterior of the closed unit ball. We prove the existence of extremal constant-sign solutions in D01,2(Ω) of the following logistic-type boundary value problem −Δu=a(x)(λu−g(u))inΩ,u=0on∂Ω=∂B(0,1),under the assumption that the nonnegative coefficient a decays like |x|−2−α with α>0, and the growth for the continuous nonlinearity g:R→R at zero and at infinity is superlinear which includes even exponential growth. We are looking for solutions in the space D01,2(Ω) which is the completion of Cc∞(Ω) with respect to the ‖∇⋅‖2,Ω-norm. For general unbounded domains in R2 including the whole plane, this completion may result in objects that do not belong to any function space. This is one of the main reasons to consider the problem in the exterior of the unit ball instead in the whole plane. Unlike in the situation of RN with N≥3, i.e. N>p=2, the behavior of the solutions in the case N=p=2 considered here is significantly different, in particular, it will be seen that the solutions are not decaying to zero at infinity, and instead are bounded away from zero. To prove our main results, new tools have to be developed here such as for example a Hopf-type lemma and a sub–supersolution method in D01,2(Ω).

A singular Monge-Ampère equation on unbounded domains

In this paper, we study the Dirichlet problem for a singular Monge-Ampere type equation on unbounded domains. For a few special kinds of unbounded convex domains,we find the explicit formulas of the solutions to the problem. For general unbounded convex domain $\Om$, we prove the existence for solutions to the problem in the space$C^{\infty}(\Om)\cap~C~(\overline{\Om})$. We also obtain the local $C^{\frac{1}{2}}$-estimate up to the $\partial~\Omega$ and the estimate for the lower bound of the solutions.

Numerical investigation of circle defining curve for two-dimensional problem with general boundaries using the scaled boundary finite element method

The scaled boundary finite element method (SBFEM) is applied to the static analysis of two dimensional elasticity problem, boundary value problems domain with the domain completely described by a circular defining curve. The scaled boundary finite element equations is formulated within a general framework integrating the influence of the distributed body force, general boundary conditions, and bounded and unbounded domain. This paper investigates the possibility of using exact geometry to form the exact description of the circular defining curve and the standard finite element shape function to approximate the defining curve. Three linear elasticity problems are presented to verify the proposed method with the analytical solution. Numerical examples show the accuracy and efficiency of the proposed method, and the performance is found to be better than using standard linear element for the approximation defining curve on the scaled boundary method.

The 3D quasilinear hyperbolic equations with nonlinear damping in a general unbounded domain

The 3D quasilinear hyperbolic equations with nonlinear damping in a general unbounded domain

Resolvent estimates of the Stokes system with Navier boundary conditions in general unbounded domains

Consider the Stokes resolvent system in general unbounded domains $\Omega \subset {\mathbb{R}^n}$, $n\geq 2$, with boundary of uniform class $C^{3}$, and Navier slip boundary condition. The main result is the resolvent estimate in function spaces of the type ${\tilde{L}^q}$ defined as $L^q\cap L^2$ when $q\geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain. As a consequence, we get that the Stokes operator generates an analytic semigroup on a solenoidal subspace ${\tilde{L}^q}_\sigma(\Omega)$ of ${\tilde{L}^q}(\Omega)$.

Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains

We consider weak solutions of the instationary Navier-Stokes system in general unbounded smooth domains $\Omega\subset \mathbb{R}^3$ and discuss several criteria to prove that the weak solution is locally or globally in time a strong solution in the sense of Serrin. Since the usual Stokes operator cannot be defined on all types of unbounded domains we have to replace the space $L^q(\Omega)$, $q>2$, by $\tilde L^q(\Omega) = L^q(\Omega) \cap L^2(\Omega)$ and Serrin's class $L^r(0,T;L^q(\Omega))$ by $L^r(0,T;\tilde L^q(\Omega))$ where $2< r <\infty$, $3< q <\infty$ and $\frac{2}{r} + \frac{3}{q} =1$.

Very weak solutions and the Fujita-Kato approach to the Navier-Stokes system in general unbounded domains

We consider the instationary Navier-Stokes system in general unbounded domains \({\Omega \subset \mathbb{R}^{n}}\), \({n\geq 3}\), with smooth boundary and construct by the Fujita-Kato method mild solutions \({u\in L^{\infty}(0,T; \tilde{L}^{n}(\Omega))}\) with initial value \({u_0\in\tilde{L}^{n}(\Omega)}\). Here the classical \({L^n(\Omega)}\)–space is replaced by \({\tilde{L}^n(\Omega)}\) where for q > 2 the space \({\tilde{L}^q}\) is defined by \({L^q\cap L^2}\). Moreover, for suitable initial values we identify mild solutions in \({L^\infty(0,T;\tilde{L}^n(\Omega))}\) with very weak solutions in Serrin’s class \({L^r (0,T;\tilde{L}^q(\Omega))}\) where \({\frac{2}{r} + \frac{n}{q} =1}\), \({2< r < \infty}\).

Very weak solutions to the Navier–Stokes system in general unbounded domains

We consider very weak instationary solutions u of the Navier–Stokes system in general unbounded domains \({\Omega \subset \mathbb{R}^n}\) , \({n \geq 3}\) , with smooth boundary, i.e., u solves the Navier–Stokes system in the sense of distributions and \({ u \in L^r (0,T;\tilde{L}^q(\Omega))}\) where \({\frac{2}{r} + \frac{n}{q} =1}\) , 2 < r < ∞. Solutions of this class have no differentiability properties and in general are not weak solutions in the sense of Leray–Hopf. However, they lie in the so-called Serrin class \({L^r(0,T;\tilde{L}^q(\Omega))}\) yielding uniqueness. To deal with the unboundedness of the domain, we work in the spaces \({\tilde{L}^q(\Omega)}\) (instead of \({L^q(\Omega)}\)) defined as \({L^q \cap L^2}\) when \({q \geq 2}\) but as Lq+ L2 when 1 < q < 2. The proofs are strongly based on duality arguments and the properties of the spaces \({\tilde{L}^q(\Omega)}\) .

Interpolation of sum and intersection spaces of L q -type and applications to the Stokes problem in general unbounded domains

In a general unbounded uniform C2-domain \({\Omega \subset \mathbb{R}^n, n \geq 3}\) , and \({1\leq q\leq \infty}\) consider the spaces \({\tilde{L}^q(\Omega)}\) defined by \({\tilde{L^q}(\Omega) := \left\{\begin{array}{ll}L^q(\Omega)+L^2(\Omega),\quad q < 2, \\ L^q(\Omega)\cap L^2(\Omega),\quad q\geq 2, \end{array}\right.}\) and corresponding subspaces of solenoidal vector fields, \({\tilde{L}^q_\sigma(\Omega)}\) . By studying the complex and real interpolation spaces of these we derive embedding properties for fractional order spaces related to the Stokes problem and Lp − Lq-type estimates for the corresponding semigroup.

Local Decay of Acoustic Waves in the Low Mach Number Limits on General Unbounded Domains Under Slip Boundary Conditions

We discuss several aspects of the problem of propagation and dispersion of acoustic waves arising in the low Mach number asymptotic limits of compressible fluid systems. A general approach is proposed based on analysis of the spectral measures associated to the corresponding wave propagator. In particular, the local decay estimates based on a result of Tosio Kato and on RAGE theorem are obtained as limit cases. The approach is applied to problems on domains their shape may vary with the Mach number.

On the energy equality of Navier–Stokes equations in general unbounded domains

We present a sufficient condition for the energy equality of Leray–Hopf’s weak solutions to the Navier–Stokes equations in general unbounded 3-dimensional domains.