Partial differential equations endowed with a Hamiltonian structure, like the Korteweg–de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability  - in fact, orbital stability, since we are...

We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $\epsilon \to 0$, the Froude number proportional to $\sqrt{\epsilon }$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $\epsilon \to 0$. The limit velocity field is solenoidal and satisfies the incompressible Oberbeck–Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we apply the spectral...

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

The isothermal Navier–Stokes–Korteweg system is used to model dynamics of a compressible fluid exhibiting phase transitions between a liquid and a vapor phase in the presence of capillarity effects close to phase boundaries. Standard numerical discretizations are known to violate discrete versions of inherent energy inequalities, thus leading to spurious dynamics of computed solutions close to static equilibria (e.g., parasitic currents). In this work, we propose a time-implicit discretization of...

In this paper we are concerned with the steady Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak, one-sided leak, velocity, vorticity, pressure and stress conditions together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. For the problem involving the static pressure and stress boundary conditions, it is proved that if the data of the problem are small enough, then there exists a solution...

We consider the steady plane flow of certain classes of viscoelastic fluids in exterior domains with a non-zero velocity prescribed at infinity. We study existence as well as asymptotic behaviour of solutions near infinity and show that for sufficiently small data the solution decays near infinity as fast as the fundamental solution to the Oseen problem.

In this paper, the axisymmetric flow in an ideal fluid outside the infinite cylinder ($r\le d$) where $(r,\theta ,z)$ denotes the cylindrical co-ordinates in ${ℝ}^3$ is considered. The motion is with swirl (i.e. the $\theta$-component of the velocity of the flow is non constant). The (non-dimensional) equation governing the phenomenon is (Pd) displayed below. It is known from e.g. that for the problem without swirl ($f_q=0$ in (f)) in the whole space, as the flux constant $k$ tends to $\infty$, 1) $\mathrm{dist}(0z,\partial A)=O\left(k^{1/2}\right)$; $\mathrm{diam}A=O(exp(-c_0{k}^{3/2}))$; 2) ${\left(k^{1/2}\Psi \right)}_{k\in \mathbb{N}}$ converges to a vortex cylinder $U_m$ (see...

Steady-state system of equations for incompressible, possibly non-Newtonean of the $p$-power type, viscous flow coupled with the heat equation is considered in a smooth bounded domain $\Omega \subset {ℝ}^n$, $n=2$ or 3, with heat sources allowed to have a natural $L^1$-structure and even to be measures. The existence of a distributional solution is shown by a fixed-point technique for sufficiently small data if $p>3/2$ (for $n=2$) or if $p>9/5$ (for $n=3$).

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at ($\eta =0,\psi =0$)).