Critical points for functionals with quasilinear singular Euler–Lagrange equations

For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ .     

Modified Euler’s method with a graded mesh for a class of Volterra integral equations with weakly singular kernel

A modified Euler’s method applied on a graded mesh is considered for numerical solution of a class of Volterra integral equations with weakly singular kernel which depends on a parameter μ > 0. It is shown that the convergence rate of the considered method is higher than those of earlier ones for the case when μ ≤ 1. The convergence rate is also obtained in the case μ > 1. Using some numerical examples, we illustrate the theoretical results.     

Sixth-order Cahn–Hilliard equations with singular nonlinear terms

Our aim in this paper is to study the well-posedness for a class of sixth-order Cahn–Hilliard equations with singular nonlinear terms. More precisely, we prove the existence and uniqueness of variational solutions, based on a variational inequality.     

From quantum Euler–Maxwell equations to incompressible Euler equations

The combined quasi-neutral and non-relativistic limit of compressible quantum Euler–Maxwell equations for plasmas is studied in this paper. For well-prepared initial data, it is shown that the smooth solution of compressible quantum Euler–Maxwell equations converges to the smooth solution of incompressible Euler equations by using the modulated energy method. Furthermore, the associated convergence rates are also obtained.     

Global Well-Posedness for Euler–Boussinesq System with Critical Dissipation

In this paper we study a fractional diffusion Boussinesq model which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion for the temperature. We prove global well-posedness results.     

On a boundary problem with Neumann’s condition for 3D singular elliptic equations

The present work is devoted to the studying of a boundary-value problem with Neumann’s condition for three-dimensional elliptic equation with singular coefficients. The main result is a proof of the unique solvability of the problem considered. An energy integral method and a Green’s function method were used as the main tools in the proof of the main result. The unique solution is found in an explicit form, which contains Appel’s hypergeometric functions.     

Improved blowup results for the Euler and Euler–Poisson equations with repulsive forces

Some results are presented on the formation of singularities in the solutions of the radially-symmetric N-dimensional Euler or Euler–Poisson equations with repulsive forces. Based on the integration method of M.W. Yuen, we generalize the blowup results with constant compact radius R   of solutions to the case with general compact radius R(t)$R(t)$ and to the case with no compact support restriction.     

Vanishing viscosity limit for Navier–Stokes equations with general viscosity to Euler equations

In this paper, we study vanishing viscosity limit of 1-D isentropic compressible Navier–Stokes equations with general viscosity to isentropic Euler equations. Firstly, we improve estimates of the entropy flux, then we obtain that the weak solution of the isentropic Euler equations is the inviscid limit of the isentropic compressible Navier–Stokes equations with general viscosity using the compensated compactness frame recently established by G.-Q. Chen and M. Perepelitsa.     

2D constrained Navier–Stokes equations

We study 2D Navier–Stokes equations with a constraint forcing the conservation of the energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier–Stokes equation on ${\mathbb{R}}^2$ and ${\mathbb{T}}^2$, by a fixed point argument. We also show that the solution of the constrained equation converges to the solution of the Euler equation as the viscosity ν vanishes.     

Nonlinear pseudoparabolic equations as singular limit of reaction–diffusion equations

In this article, a solution of a nonlinear pseudoparabolic equation is constructed as a singular limit of a sequence of solutions of quasilinear hyperbolic equations. If a system with cross diffusion, modelling the reaction and diffusion of two biological, chemical, or physical substances, is reduced then such an hyperbolic equation is obtained. For regular solutions even uniqueness can be shown, although the needed regularity can only be proved in two dimensions.     

On Global Attractor for 2D Kirchhoff–Boussinesq Model with Supercritical Nonlinearity

Dynamics for a class of nonlinear 2D Kirchhoff–Boussinesq models is studied. These nonlinear plate models are characterized by the presence of a nonlinear source that alone leads to finite-time blow up of solutions. In order to counteract, restorative forces are introduced, which however are of a supercritical nature. This raises natural questions such as: (i) wellposedness of finite energy (weak) solutions, (ii) their regularity, and (iii) long time behavior of both weak and strong solutions. It is shown that finite energy solutions do exist globally, are unique and satisfy Hadamard wellposedness criterium. In addition, weak solutions corresponding to “strong” initial data (i.e., strong solutions) enjoy, likewise, the full Hadamard wellposedness. The proof is based on logarithmic control of the lack of Sobolev's embedding. In addition to wellposedness, long time behavior is analyzed. Viscous damping added to the model controls long time behaviour of solutions. It is shown that both weak and (resp. strong) solutions admit compact global attractors in the finite energy norm, (resp. strong topology of strong solutions). The proof of long time behavior is based on Ball's method [2 Ball , J. ( 2004 ). Global attractors for semilinear wave equations . Discr. Cont. Dyn. Sys. 10 : 31 – 52 .[Crossref], [Web of Science ®] , [Google Scholar]] and on recent asymptotic quasi-stability inequalities established in [11 Chueshov , I. , Lasiecka , I. ( 2008 ). Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping . Memoirs of the American Mathematical Society, Vol. 912 . Providence , RI : American Mathematical Society .[Crossref] , [Google Scholar]]. These inequalities enable to prove that strong attractors are finite-dimensional and the corresponding trajectories can exhibit C ^∞ smoothness.     

Solvability of the 3D rotating Navier–Stokes equations coupled with a 2D biharmonic problem with obstacles and gradient restriction

A coupled system by the 3D rotating Navier–Stokes equations with a mixed boundary condition and a 2D biharmonic problem with two obstacles and the gradient restriction is investigated in this paper. Using the Schauder’s fixed point theorem, we show the existence of a strong solution for a sufficiently large viscosity ν and sufficiently small data.     

On soliton solutions for Boussinesq–Burgers equations

The periodic wave solutions for Boussinesq–Burgers equations are obtained by using of Jacobi elliptic function method, in the limit cases, the multiple soliton solutions are also obtained. The properties of some periodic and soliton solution for this system are shown by some figures.     

Fractional variational integrators for fractional Euler–Lagrange equations with holonomic constraints

In this paper, the fractional variational integrators developed by Wang and Xiao (2012) [28] are extended to the fractional Euler–Lagrange (E–L) equations with holonomic constraints. The corresponding fractional discrete E–L equations are derived, and their local convergence is discussed. Some fractional variational integrators are presented. The suggested methods are shown to be efficient by some numerical examples.     

Hyperbolic–parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation

We consider the hyperbolic?Cparabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t ?? +???. The case where b(t) ~ (1?+?t)^?p with p?<?1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t ???+??? and as ${\varepsilon \to 0^{+}}$ . In this paper we show that these results cannot be true for p?> 1, but they remain true in the critical case p?=?1.     

For the vorticity–velocity–pressure form of the Navier–Stokes equations on a bounded plane domain with corners

We show existence and regularity for the boundary value problems of the Navier–Stokes equations with non-standard BCs on a bounded plane domain with non-convex corners. We assign the vorticity value $\omega ={\omega }_0$ and the velocity normal component $u?n=u_0?n$ over the non-convex corner, the dynamic pressure value $p+{\left|u\right|}^2/2=p_0$ over inflow and outflow boundaries, and so on. We construct the corner singularity functions for the Stokes operator with zero vorticity and velocity normal component BCs, subtract its leading singularity from the solution by defining the coefficient of the singularity and show increased regularity for the remainder. The solution is determined by the smoother part and the coefficients of the singularities. It is seen from the singularity that the dynamic pressure has a transition layer that changes the sign (at $\theta =\pi /2$ in the domain). The obtained results can be applied to general polygonal domains and the cavity flows.