# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

September 2021 , Volume 20 , Issue 9

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2021, 20(9): 2885-2914 doi: 10.3934/cpaa.2021068 +[Abstract](451) +[HTML](173) +[PDF](632.34KB)
Abstract:

We study the theory of relativistic viscous hydrodynamics introduced in [14, 58], which provided a causal and stable first-order theory of relativistic fluids with viscosity in the case of barotropic fluids. The local well-posedness of its equations of motion has been previously established in Gevrey spaces. Here, we improve this result by proving local well-posedness in Sobolev spaces.

2021, 20(9): 2915-2939 doi: 10.3934/cpaa.2021071 +[Abstract](446) +[HTML](180) +[PDF](484.58KB)
Abstract:

We show that the operator

is a Calderon-Zygmund operator. Here for \begin{document}$K \in L^\infty( \mathbb{R}^n \times \mathbb{R}^n)$\end{document}, and \begin{document}$s,s_1,s_2 \in (0,1)$\end{document} with \begin{document}$s_1+s_2 = 2s$\end{document} we have

This operator is motivated by the recent work [12] where it appeared as analogue of the Riesz transforms for the equation

2021, 20(9): 2941-2963 doi: 10.3934/cpaa.2021090 +[Abstract](413) +[HTML](170) +[PDF](759.85KB)
Abstract:

We study the motion of the negative curved symmetric two and three center problem on the Poincaré upper semi plane model for a surface of constant negative curvature \begin{document}$\kappa$\end{document}, which without loss of generality we assume \begin{document}$\kappa = -1$\end{document}. Using this model, we first derive the equations of motion for the \begin{document}$2$\end{document}-and \begin{document}$3$\end{document}-center problems. We prove that for \begin{document}$2$\end{document}–center problem, there exists a unique equilibrium point and we study the dynamics around it. For the motion restricted to the invariant \begin{document}$y$\end{document}–axis, we prove that it is a center, but for the general two center problem it is unstable. For the \begin{document}$3$\end{document}–center problem, we show the non-existence of equilibrium points. We study two particular integrable cases, first when the motion of the free particle is restricted to the \begin{document}$y$\end{document}–axis, and second when all particles are along the same geodesic. We classify the singularities of the problem and introduce a local and a global regularization of all them. We show some numerical simulations for each situation.

2021, 20(9): 2965-2989 doi: 10.3934/cpaa.2021091 +[Abstract](373) +[HTML](159) +[PDF](505.71KB)
Abstract:

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in \begin{document}$L^2$\end{document}-based Sobolev spaces \begin{document}$H^s$\end{document} and \begin{document}$H^l$\end{document} for the electromagnetic field \begin{document}$\phi$\end{document} and the potential \begin{document}$A$\end{document}, respectively, the minimal regularity assumptions are \begin{document}$s > \frac{1}{2}$\end{document} and \begin{document}$l > \frac{1}{4}$\end{document}, which leaves a gap of \begin{document}$\frac{1}{2}$\end{document} and \begin{document}$\frac{1}{4}$\end{document} to the critical regularity with respect to scaling \begin{document}$s_c = l_c = 0$\end{document}. This gap can be reduced for data in Fourier-Lebesgue spaces \begin{document}$\widehat{H}^{s, r}$\end{document} and \begin{document}$\widehat{H}^{l, r}$\end{document} to \begin{document}$s> \frac{21}{16}$\end{document} and \begin{document}$l > \frac{9}{8}$\end{document} for \begin{document}$r$\end{document} close to \begin{document}$1$\end{document}, whereas the critical exponents with respect to scaling fulfill \begin{document}$s_c \to 1$\end{document}, \begin{document}$l_c \to 1$\end{document} as \begin{document}$r \to 1$\end{document}. Here \begin{document}$\|f\|_{\widehat{H}^{s, r}} : = \| \langle \xi \rangle^s \tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 < r \le 2 \, , \, \frac{1}{r}+\frac{1}{r'} = 1 \, .$\end{document} Thus the gap is reduced for \begin{document}$\phi$\end{document} as well as \begin{document}$A$\end{document} in both gauges.

2021, 20(9): 2991-3028 doi: 10.3934/cpaa.2021092 +[Abstract](390) +[HTML](166) +[PDF](615.94KB)
Abstract:

The purpose of this paper is to investigate the stabilization of a locally coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combined with the multiplier method, we prove a polynomial energy decay rate of order \begin{document}$t^{-1}$\end{document}.

2021, 20(9): 3029-3063 doi: 10.3934/cpaa.2021095 +[Abstract](394) +[HTML](173) +[PDF](578.61KB)
Abstract:

We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity–Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.

2021, 20(9): 3065-3092 doi: 10.3934/cpaa.2021096 +[Abstract](406) +[HTML](158) +[PDF](520.71KB)
Abstract:

The aim of this paper is to study the multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions and concave-convex nonlinearities with subcritical or critical growth. Applying Nehari manifold, fibering maps and Ljusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the global maximum set of \begin{document}$K$\end{document}.

2021, 20(9): 3093-3111 doi: 10.3934/cpaa.2021097 +[Abstract](394) +[HTML](172) +[PDF](690.71KB)
Abstract:

The leading-order approximation to a Filippov system \begin{document}$f$\end{document} about a generic boundary equilibrium \begin{document}$x^*$\end{document} is a system \begin{document}$F$\end{document} that is affine one side of the boundary and constant on the other side. We prove \begin{document}$x^*$\end{document} is exponentially stable for \begin{document}$f$\end{document} if and only if it is exponentially stable for \begin{document}$F$\end{document} when the constant component of \begin{document}$F$\end{document} is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for \begin{document}$F$\end{document}. We also show exponential stability is preserved under small perturbations to the pieces of \begin{document}$F$\end{document}. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of \begin{document}$F$\end{document} have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of \begin{document}$x^*$\end{document} from the general Filippov system \begin{document}$f$\end{document} to the simpler system \begin{document}$F$\end{document}. Yet in general this problem remains difficult. We provide a four-dimensional example of \begin{document}$F$\end{document} for which orbits appear to converge to \begin{document}$x^*$\end{document} in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of \begin{document}$F$\end{document} can in this case be reduced to the combination of a one-dimensional return map and a scalar function.

2021, 20(9): 3113-3128 doi: 10.3934/cpaa.2021098 +[Abstract](355) +[HTML](160) +[PDF](397.75KB)
Abstract:

In this work, we are concerned with the problem of boundary exponential stabilization, in a Hilbert space \begin{document}$H$\end{document}, of parabolic type equations, namely equations for which their linear parts generate analytic \begin{document}$C_0-$\end{document}semigroups. We consider the case where the projection of the linear leading operator, on a given Riesz basis of \begin{document}$H$\end{document}, is non-diagonal. We do not assume that the linear operator has compact resolvent. Therefore, the Riesz basis is not necessarily an eigenbasis. The boundary stabilizer is given in a simple linear feedback form, of finite-dimensional structure, involving only the Riesz basis. To illustrate the results, at the end of the paper, we provide an example of stabilization of a fourth-order evolution equation on the half-axis.

2021, 20(9): 3129-3142 doi: 10.3934/cpaa.2021099 +[Abstract](313) +[HTML](131) +[PDF](413.25KB)
Abstract:

By comparing the original equations with the corresponding stationary ones, the moderate deviation principle (MDP) is established for unbounded additive functionals of several different models of distribution dependent SDEs, with non-degenerate and degenerate noises.

2021, 20(9): 3143-3159 doi: 10.3934/cpaa.2021100 +[Abstract](299) +[HTML](152) +[PDF](417.72KB)
Abstract:

In this paper, we establish uniform \begin{document}$W^{1,p}$\end{document} estimates for composite material problems which can be described by a divergence form elliptic system on a nonsmooth domain composed of a finite number of subdomains. We want to derive global \begin{document}$W^{1,p}$\end{document} regularity under the assumption that the coefficients are almost \begin{document}$(\delta,R)$\end{document}-vanishing of codimension 1 (see Definition 1.2) in each of multiple subdomains and the boundaries of subdomains are Reifenberg flat, moreover the estimates do not depend on the distance between these subdomains.

2021, 20(9): 3161-3192 doi: 10.3934/cpaa.2021101 +[Abstract](297) +[HTML](145) +[PDF](562.05KB)
Abstract:

In this paper we use a potential-theoretic approach to establish various representation theorems and Poisson-Jensen-type formulas for subharmonic functions in sub-Riemannian settings. We also characterize the Radon measures in \begin{document}$\mathbb{R}^N$\end{document} which are the Riesz-measures of bounded-above subharmonic functions in the whole space \begin{document}$\mathbb{R}^N$\end{document}.

2021, 20(9): 3193-3213 doi: 10.3934/cpaa.2021102 +[Abstract](332) +[HTML](165) +[PDF](456.44KB)
Abstract:

In this work we study, by a semigroup approach, a transmission problem based on biharmonic equations with boundary and transmission conditions, in two juxtaposed habitats. We give a result of existence and uniqueness of the classical solution in \begin{document}$L^p$\end{document}-spaces, for \begin{document}$p \in (1,+\infty)$\end{document}, using analytic semigroups and operators sum theory in Banach spaces. To this end, we invert explicitly the determinant operator of the transmission system in \begin{document}$L^p$\end{document}-spaces using the \begin{document}$\mathcal{E}_{\infty}$\end{document}-calculus and the Dore-Venni sums theory.

2021, 20(9): 3215-3234 doi: 10.3934/cpaa.2021103 +[Abstract](389) +[HTML](170) +[PDF](470.81KB)
Abstract:

In this paper, we first study the fractional Yamabe solitons, which are the self-similar solutions to fractional Yamabe flow.We prove some rigidity results and Liouville type results for such solitons.We thenconsider the fractional Nirenberg problem:the problem of prescribing fractional order curvature on the sphere.More precisely, we prove that there exists a conformal metric on the unit sphere such that itsfractional order curvature is \begin{document}$f$\end{document}, when \begin{document}$f$\end{document} possesses certain reflection or rotation symmetry.

2021, 20(9): 3235-3258 doi: 10.3934/cpaa.2021104 +[Abstract](354) +[HTML](140) +[PDF](471.52KB)
Abstract:

The phenomena of concentration and cavitation for the Riemann problem of the Baer-Nunziato (BN) two-phase flow model has been investigated in this paper. By using the characteristic analysis method, the formation of \begin{document}$\delta-$\end{document}waves and vacuum states are obtained as the pressure for both phases vanish in the BN model. The solid contact wave is carefully dealt. The comparison with the solutions of pressureless two-phase model shows that, two shock waves tend to a \begin{document}$\delta-$\end{document}shock solution, and two rarefaction waves tend to a two contact discontinuity solution when the solid contact discontinuity is involved. Moreover, the detailed Riemann solutions for two-phase flow model are given as the double pressure parameters vanish. This may contribute to the design of numerical schemes in the future research.

2021, 20(9): 3259-3297 doi: 10.3934/cpaa.2021105 +[Abstract](525) +[HTML](153) +[PDF](706.41KB)
Abstract:

Let \begin{document}$k,N\in \mathbb{N}$\end{document} with \begin{document}$1\le k\le N$\end{document} and let \begin{document}$\Omega = \Omega_1 \times \Omega_2$\end{document} be an open set in \begin{document}$\mathbb{R}^k \times \mathbb{R}^{N-k}$\end{document}. For \begin{document}$p\in (1,\infty)$\end{document} and \begin{document}$q \in (0,\infty),$\end{document} we consider the following weighted Sobolev type inequality:

for some \begin{document}$C>0$\end{document}. Depending on the values of \begin{document}$N,k,p,q$\end{document} we have identified various pairs of Lorentz spaces, Lorentz-Zygmund spaces and weighted Lebesgue spaces for \begin{document}$(g_1, g_2)$\end{document} so that (0.1) holds. Furthermore, we give a sufficient condition on \begin{document}$g_1,g_2$\end{document} so that the best constant in (0.1) is attained in the Beppo-Levi space \begin{document}$\mathcal{D}^{1,p}_0(\Omega)$\end{document}-the completion of \begin{document}$\mathcal{C}^1_c(\Omega)$\end{document} with respect to \begin{document}$\|\nabla u\|_{L p(\Omega)}$\end{document}.

2020 Impact Factor: 1.916
5 Year Impact Factor: 1.510
2020 CiteScore: 1.9