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Communications on Pure & Applied Analysis

March 2019 , Volume 18 , Issue 2

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On the one-dimensional continuity equation with a nearly incompressible vector field
Nikolay A. Gusev
2019, 18(2): 559-568 doi: 10.3934/cpaa.2019028 +[Abstract](40) +[HTML](42) +[PDF](315.02KB)

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field \begin{document} $b\colon (0,T) × \mathbb{R}^d \to \mathbb{R}^d$ \end{document}, \begin{document} $T>0$ \end{document}. This class of vector fields arises in the context of hyperbolic conservation laws (in particular, the Keyfitz-Kranzer system, which has applications in nonlinear elasticity theory).

It is well known that in the generic multi-dimensional case (\begin{document}$d≥ 1$\end{document}) near incompressibility is sufficient for existence of bounded weak solutions, but uniqueness may fail (even when the vector field is divergence-free), and hence further assumptions on the regularity of \begin{document} $b$ \end{document} (e.g. Sobolev regularity) are needed in order to obtain uniqueness.

We prove that in the one-dimensional case (\begin{document}$d = 1$\end{document}) near incompressibility is sufficient for existence and uniqueness of locally integrable weak solutions. We also study compactness properties of the associated Lagrangian flows.

Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition
Xumin Gu
2019, 18(2): 569-602 doi: 10.3934/cpaa.2019029 +[Abstract](27) +[HTML](22) +[PDF](443.89KB)

We consider a free boundary problem for the axially symmetric incompressible ideal magnetohydrodynamic equations that describe the motion of the plasma in vacuum. Both the plasma magnetic field and vacuum magnetic field are tangent along the plasma-vacuum interface. Moreover, the vacuum magnetic field is composed in a non-simply connected domain and hence is non-trivial. Under the non-collinearity condition for the plasma and vacuum magnetic fields, we prove the local well-posedness of the problem in Sobolev spaces.

Well-posedness issues for some critical coupled non-linear Klein-Gordon equations
Radhia Ghanmi and Tarek Saanouni
2019, 18(2): 603-623 doi: 10.3934/cpaa.2019030 +[Abstract](31) +[HTML](17) +[PDF](428.56KB)

The initial value problem for some coupled non-linear wave equations is investigated. In the defocusing case, global well-posedness and ill-posedness results are obtained. In the focusing sign, the existence of global and non global solutions are discussed via the potential-well theory. Finally, strong instability of standing waves are established.

On Positive solutions of integral equations with the weighted Bessel potentials
Mingchun Wang, Jiankai Xu and Huoxiong Wu
2019, 18(2): 625-641 doi: 10.3934/cpaa.2019031 +[Abstract](21) +[HTML](26) +[PDF](375.94KB)

This paper is devoted to exploring the properties of positive solutions for a class of nonlinear integral equation(s) involving the Bessel potentials, which are equivalent to certain partial differential equations under appropriate integrability conditions. With the help of regularity lifting theorem, we obtain an integrability interval of positive solutions and then extend the integrability interval to the whole [1, ∞) by the properties of the Bessel kernels and some delicate analysis techniques. Meanwhile, the radial symmetry and the sharp exponential decay of positive solutions are also obtained. Furthermore, as an application, we establish the uniqueness theorem of the corresponding partial differential equations.

Continuous data assimilation for the 3D primitive equations of the ocean
Yuan Pei
2019, 18(2): 643-661 doi: 10.3934/cpaa.2019032 +[Abstract](17) +[HTML](17) +[PDF](353.23KB)

In this article, we show that the continuous data assimilation algorithm is valid for the 3D primitive equations of the ocean. Namely, the assimilated solution converges to the reference solution in $L^2$ norm at an exponential rate in time. We also prove the global existence of strong solution to the assimilated system.

Infinite energy solutions for the (3+1)-dimensional Yang-Mills equation in Lorenz gauge
Hartmut Pecher
2019, 18(2): 663-688 doi: 10.3934/cpaa.2019033 +[Abstract](24) +[HTML](22) +[PDF](420.5KB)

We prove that the Yang-Mills equation in Lorenz gauge in the (3+1)-dimensional case is locally well-posed for data of the gauge potential in \begin{document} $H^s$ \end{document} and the curvature in \begin{document} $H^r$ \end{document}, where \begin{document} $s >\frac{5}{7}$ \end{document} and \begin{document} $r > -\frac{1}{7}$ \end{document}, respectively. This improves a result by Tesfahun [16]. The proof is based on the fundamental results of Klainerman-Selberg [6] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [14] and Tesfahun [16].

On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation
Van Duong Dinh
2019, 18(2): 689-708 doi: 10.3934/cpaa.2019034 +[Abstract](26) +[HTML](23) +[PDF](392.3KB)

In this paper we study dynamical properties of blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. We establish a profile decomposition and a compactness lemma related to the equation. As a result, we obtain the \begin{document}$L^2$\end{document}-concentration and the limiting profile with minimal mass of blow-up solutions.

Weak solutions to stationary equations of heat transfer in a magnetic fluid
Youcef Amirat and Kamel Hamdache
2019, 18(2): 709-734 doi: 10.3934/cpaa.2019035 +[Abstract](30) +[HTML](35) +[PDF](410.83KB)

We consider the differential system describing the stationary heat transfer in a magnetic fluid in the presence of a heat source and an external magnetic field. The system consists of the stationary incompressible Navier-Stokes equations, the magnetostatic equations and the stationary heat equation. We prove, for the differential system posed in a bounded domain of \begin{document} $\mathbb{R}^3$ \end{document} and equipped with Fourier boundary conditions, the existence of weak solutions by using a regularization of the Kelvin force and the thermal power.

Steady flows of an Oldroyd fluid with threshold slip
Evgenii S. Baranovskii
2019, 18(2): 735-750 doi: 10.3934/cpaa.2019036 +[Abstract](24) +[HTML](27) +[PDF](334.08KB)

We consider a mathematical model that describes 3D steady flows of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain under mixed boundary conditions, including a threshold-slip boundary condition. Using the concept of weak solutions, we reduce the original slip problem to a coupled system of variational inequalities and equations for the velocity field and stresses. For arbitrary large data (forcing and boundary data) and suitable material constants, we prove the existence of weak solutions and establish some of their properties.

Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities
Xin Liu
2019, 18(2): 751-794 doi: 10.3934/cpaa.2019037 +[Abstract](32) +[HTML](25) +[PDF](461.38KB)

This work is devoted to studying the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such a scenario, the boundary no longer provides friction and therefore the perturbation of the angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the stability of the uniformly rotating solutions with a small angular velocity. In particular, the initial perturbation which preserves the angular momentum will be stable in the sense that the global strong solution to the Navier-Stokes equations exists and the perturbation is uniformly bounded and small in time.

Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution
Hyun-Jung Kim
2019, 18(2): 795-807 doi: 10.3934/cpaa.2019038 +[Abstract](19) +[HTML](43) +[PDF](322.9KB)

A mild formulation for stochastic parabolic Anderson model with time-homogeneous Gaussian potential suggests a way of defining a solution to obtain its optimal regularity. Two different interpretations in the equation or in the mild formulation are possible with usual pathwise product and the Wick product: the usual pathwise interpretation is mainly discussed. We emphasize that a modified version of parabolic Schauder estimates is a key idea for the existence and uniqueness of a mild solution. In particular, the mild formulation is crucial to investigate a relation between the equations with usual pathwise product and the Wick product.

Long term behavior of a random Hopfield neural lattice model
Xiaoying Han, Peter E. Kloeden and Basiru Usman
2019, 18(2): 809-824 doi: 10.3934/cpaa.2019039 +[Abstract](21) +[HTML](45) +[PDF](356.86KB)

A Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of square summable bi-infinite sequences. Then the existence and uniqueness of solutions, as well as long term dynamics of solutions are investigated.

Attractors of the strongly damped Kirchhoff wave equation on $\mathbb{R}^{N}$
Pengyan Ding and Zhijian Yang
2019, 18(2): 825-843 doi: 10.3934/cpaa.2019040 +[Abstract](22) +[HTML](69) +[PDF](394.02KB)

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on \begin{document}$\mathbb{R}^N$\end{document}: \begin{document}$u_{tt}-φ(x)Δ u_{t}-φ(x)M(\|\nabla u\|^{2})Δ u+f(u) = h(x)$\end{document}. It proves that when the growth exponent \begin{document}$p$\end{document} of the nonlinearity \begin{document}$f(u) $\end{document} is up to the supercritical range: \begin{document}$ 1≤ p < p^{**}(\equiv \frac{N+4}{(N-4)^+})$\end{document}, the related solution semigroup has in weighted energy space a (strong) global attractor and a partially strong exponential attractor, respectively. In particular, the partially strong exponential attractor becomes the strong one in non-supercritical case (i.e., \begin{document}$1≤ p≤ p^{*}(\equiv \frac{N+2}{N-2})$\end{document}).

An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation
Éder Rítis Aragão Costa
2019, 18(2): 845-868 doi: 10.3934/cpaa.2019041 +[Abstract](77) +[HTML](75) +[PDF](406.96KB)

In this paper we prove versions, in Fréchet spaces, of the classical theorems related to exponential dichotomy for a sequence of continuous linear operators on Banach spaces. To be more specific, here we define a kind of exponential dichotomy in Fréchet spaces, which extends the former one in Banach spaces, establish necessary conditions for its existence and provide sufficient conditions for its stability under perturbation.

We apply the conclusions by providing an example of a semigroup of bounded linear operators, on a Fréchet space, which has this new exponential dichotomy but does not in Banach spaces, namely, \begin{document}$\{e^{mΔ}:\; m∈ \mathbb{N}\}$\end{document}, where \begin{document}$Δ$\end{document} is the Laplace operator on the unbounded domain \begin{document}$\mathbb{R}^{n}\setminus \{0\}$\end{document}.

Also, we show how these new concepts allow us to study a hyperbolic equilibrium point of a backwards heat equation with nonlinearity involving convolution products, which cannot be obtained from the knowledge of exponential dichotomy in Banach spaces.

A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator
Ismail Kombe and Abdullah Yener
2019, 18(2): 869-886 doi: 10.3934/cpaa.2019042 +[Abstract](28) +[HTML](38) +[PDF](369.87KB)

In this paper we exhibit some sufficient conditions that imply general weighted \begin{document}$L^{p}$\end{document} Rellich type inequality related to Greiner operator without assuming a priori symmetric hypotheses on the weights. More precisely, we prove that given two nonnegative functions \begin{document}$a$\end{document} and \begin{document}$b$\end{document}, if there exists a positive supersolution \begin{document}$\vartheta $\end{document} of the Greiner operator \begin{document}$Δ _{k}$\end{document} such that

almost everywhere in \begin{document}$\mathbb{R}^{2n+1}, $\end{document} then \begin{document}$a$\end{document} and \begin{document}$b$\end{document} satisfy a weighted \begin{document}$L^{p}$\end{document} Rellich type inequality. Here, \begin{document}$p>1$\end{document} and \begin{document}$Δ _{k} = \sum\nolimits_{j = 1}^n {} \left(X_{j}^{2}+Y_{j}^{2}\right) $\end{document} is the sub-elliptic operator generated by the Greiner vector fields

where \begin{document}$\left( z, l\right) = \left( x, y, l\right) ∈\mathbb{R}^{2n+1} = \mathbb{R}^{n}×\mathbb{R}^{n}×\mathbb{R}, $\end{document} \begin{document}$|z{\rm{|}} = \sqrt {\sum\nolimits_{j = 1}^n {} \left( {x_j^2 + y_j^2} \right)} $\end{document} and \begin{document}$k≥ 1$\end{document}. The method we use is quite practical and constructive to obtain both known and new weighted Rellich type inequalities. On the other hand, we also establish a sharp weighted \begin{document}$L^{p}$\end{document} Rellich type inequality that connects first to second order derivatives and several improved versions of two-weight \begin{document}$L^{p}$\end{document} Rellich type inequalities associated to the Greiner operator \begin{document}$Δ _{k}$\end{document} on smooth bounded domains \begin{document}$Ω $\end{document} in \begin{document}$\mathbb{R}^{2n+1}$\end{document}.

Vanishing viscosity limit of 1d quasilinear parabolic equation with multiple boundary layers
Jing Wang and Lining Tong
2019, 18(2): 887-910 doi: 10.3934/cpaa.2019043 +[Abstract](23) +[HTML](16) +[PDF](351.82KB)

In this paper, we study the limiting behavior of solutions to a 1D two-point boundary value problem for viscous conservation laws with genuinely-nonlinear fluxes as \begin{document}$\varepsilon$\end{document} goes to zero. We here discuss different types of non-characteristic boundary layers occurring on both sides. We first construct formally the three-term approximate solutions by using the method of matched asymptotic expansions. Next, by energy method we prove that the boundary layers are nonlinearly stable and thus it is proved the boundary layer effects are just localized near both boundaries. Consequently, the viscous solutions converge to the smooth inviscid solution uniformly away from the boundaries. The rate of convergence in viscosity is optimal.

Attractors and their stability on Boussinesq type equations with gentle dissipation
Zhijian Yang, Pengyan Ding and Xiaobin Liu
2019, 18(2): 911-930 doi: 10.3934/cpaa.2019044 +[Abstract](23) +[HTML](17) +[PDF](403.74KB)

The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:\begin{document}$ u_{tt}+Δ^2 u+(-Δ)^{α} u_{t}-Δ f(u) = g(x)$\end{document}, with \begin{document}$α∈ (0, 1)$\end{document}. For general bounded domain \begin{document}$Ω\subset \mathbb{R}^N (N≥1)$\end{document}, we show that there exists a critical exponent \begin{document}$p_α\equiv\frac{N+2(2α-1)}{(N-2)^+}$\end{document} depending on the dissipative index α such that when the growth p of the nonlinearity f(u) is up to the range: \begin{document}$1≤p <p_α$\end{document}, (ⅰ) the weak solutions of the equations are of additionally global smoothness when \begin{document}$t>0$\end{document}; (ⅱ) the related dynamical system possesses a global attractor \begin{document}$\mathcal{A}_α$\end{document} and an exponential attractor \begin{document}$\mathcal{A}^α_{exp}$\end{document} in natural energy space for each \begin{document}$α∈ (0, 1)$\end{document}, respectively; (ⅲ) the family of global attractors \begin{document}$\{\mathcal{A}_α\}$\end{document} is upper semicontinuous at each point \begin{document}$α_0∈ (0,1] $\end{document}, i.e., for any neighborhood U of \begin{document}$\mathcal{A}_{α_0}, \mathcal{A}_α\subset U$\end{document} when \begin{document}$|α-α_0|\ll 1$\end{document}. These results extend those for structural damping case: \begin{document} $α∈ [1, 2)$\end{document} in [31,32].

Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation
Anupam Sen and T. Raja Sekhar
2019, 18(2): 931-942 doi: 10.3934/cpaa.2019045 +[Abstract](51) +[HTML](25) +[PDF](426.59KB)

In this article, we study the Riemann problem for a strictly hyperbolic system of conservation laws under the linear approximation of flux functions with three parameters. The approximation does not affect the structure of Riemann problem. Furthermore, we prove that the Riemann solution to the approximated system converges to the original system as the perturbation parameter tends to zero.

Semi-hyperbolic patches of solutions to the two-dimensional compressible magnetohydrodynamic equations
Jianjun Chen and Geng Lai
2019, 18(2): 943-958 doi: 10.3934/cpaa.2019046 +[Abstract](24) +[HTML](20) +[PDF](368.72KB)

We construct semi-hyperbolic patches of solutions, in which one family out of two families of wave characteristics start on sonic curves and end on transonic shock waves, to the two-dimensional (2D) compressible magnetohydrodynamic (MHD) equations. This type of flow patches appear frequently in transonic flow problems. In order to use the method of characteristic decomposition to construct such a flow patch, we also derive a group of characteristic decompositions for 2D self-similar MHD equations.

Fractal analysis of canard cycles with two breaking parameters and applications
Renato Huzak and Domagoj Vlah
2019, 18(2): 959-975 doi: 10.3934/cpaa.2019047 +[Abstract](28) +[HTML](21) +[PDF](608.78KB)

In previous work [13] we introduced a new box dimension method for computation of the number of limit cycles in planar slow-fast systems, Hausdorff close to balanced canard cycles with one breaking mechanism (the Hopf breaking mechanism or the jump breaking mechanism). This geometric approach consists of a simple iteration method for finding one orbit of the so-called slow relation function and of the calculation of the box dimension of that orbit. Then we read the cyclicity of the balanced canard cycles from the box dimension. The purpose of the present paper is twofold. First, we generalize the box dimension method to canard cycles with two breaking mechanisms. Second, we apply the method from [13] and our generalized method to a number of interesting examples of canard cycles with one breaking mechanism and with two breaking mechanisms respectively.

A nonlinear model for marble sulphation including surface rugosity: Theoretical and numerical results
Elena Bonetti, Cecilia Cavaterra, Francesco Freddi, Maurizio Grasselli and Roberto Natalini
2019, 18(2): 977-998 doi: 10.3934/cpaa.2019048 +[Abstract](26) +[HTML](20) +[PDF](863.66KB)

We consider an evolution system describing the phenomenon of marble sulphation of a monument, accounting of the surface rugosity. We first prove a local in time well posedness result. Then, stronger assumptions on the data allow us to establish the existence of a global in time solution. Finally, we perform some numerical simulations that illustrate the main feature of the proposed model.

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