ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

March 2019 , Volume 18 , Issue 2

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**Abstract:**

We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field

It is well known that in the generic multi-dimensional case (

We prove that in the one-dimensional case (*and* uniqueness of *locally integrable* weak solutions. We also study compactness properties of the associated Lagrangian flows.

*+*[Abstract](27)

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**Abstract:**

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.

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**Abstract:**

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.

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**Abstract:**

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.

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**Abstract:**

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.

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**Abstract:**

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

*+*[Abstract](26)

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**Abstract:**

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

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**Abstract:**

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

*+*[Abstract](24)

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**Abstract:**

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.

*+*[Abstract](32)

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**Abstract:**

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.

*+*[Abstract](19)

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**Abstract:**

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.

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**Abstract:**

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.

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**Abstract:**

The paper investigates the existence of global and exponential attractors for the strongly damped Kirchhoff wave equation with supercritical nonlinearity on

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**Abstract:**

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,

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.

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**Abstract:**

In this paper we exhibit some sufficient conditions that imply general weighted

almost everywhere in

where

*+*[Abstract](23)

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**Abstract:**

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

*+*[Abstract](23)

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**Abstract:**

The paper investigates longtime dynamics of Boussinesq type equations with gentle dissipation:*α* such that when the growth *p* of the nonlinearity *f*(*u*) is up to the range: *U* of

*+*[Abstract](51)

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**Abstract:**

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.

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**Abstract:**

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.

*+*[Abstract](28)

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**Abstract:**

In previous work [

*+*[Abstract](26)

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**Abstract:**

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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