The authors prove the global existence and exponential stability of solutions of the given system of equations under the condition that the initial velocities and the external forces are small and the initial density is not far from a constant one. If the external forces are periodic, then solutions periodic with the same period are obtained. The investigated system of equations is a bit non-standard - for example the displacement current in the Maxwell equations is not neglected.

We discuss the validity of the Helmholtz decomposition of the Muckenhoupt $A_p$-weighted $L^p$-space ${\left(L_w^p\left(\Omega \right)\right)}^n$ for any domain $\Omega$ in ${ℝ}^n$, $n\in \mathbb{Z}$, $n\ge 2$, $1<p<\infty$ and Muckenhoupt $A_p$-weight $w\in A_p$. Set $p^{\text{'}}:=p/(p-1)$ and $w^{\text{'}}:=w^{-1/(p-1)}$. Then the Helmholtz decomposition of ${\left(L_w^p\left(\Omega \right)\right)}^n$ and ${\left(L_{w^{\text{'}}}^{p^{\text{'}}}\left(\Omega \right)\right)}^n$ and the variational estimate of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ are equivalent. Furthermore, we can replace $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ by $L_{w,\sigma }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\sigma }^{p^{\text{'}}}\left(\Omega \right)$, respectively. The proof is based on the reflexivity and orthogonality of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w,\sigma }^p\left(\Omega \right)$ and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation theorem with...

We are concerned with convergence of spectral method for the numerical solution of the initial-boundary value problem associated to the Korteweg-de Vries-Kawahara equation (Kawahara equation, in short), which is a transport equation perturbed by dispersive terms of the 3rd and 5th order. This equation appears in several fluid dynamics problems. It describes the evolution of small but finite amplitude long waves in various problems in fluid dynamics. These equations are discretized in space by the...

Space-time approximations of the FitzHugh–Nagumo system of coupled semi-linear parabolic PDEs are examined. The schemes under consideration are discontinuous in time but conforming in space and of arbitrary order. Stability estimates are presented in the natural energy norms and at arbitrary times, under minimal regularity assumptions. Space-time error estimates of arbitrary order are derived, provided that the natural parabolic regularity is present. Various physical parameters appearing in the...

Space-time approximations of the FitzHugh–Nagumo system of coupled semi-linear parabolic PDEs are examined. The schemes under consideration are discontinuous in time but conforming in space and of arbitrary order. Stability estimates are presented in the natural energy norms and at arbitrary times, under minimal regularity assumptions. Space-time error estimates of arbitrary order are derived, provided that the natural parabolic regularity is present....

The Coupled Cluster (CC) method is a widely used and highly successful high precision method for the solution of the stationary electronic Schrödinger equation, with its practical convergence properties being similar to that of a corresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method been analyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in [Schneider, 2009]. Recently, we globalized the CC formulation to the full continuous space, giving a root...

The incompressible Navier-Stokes problem is discretized in time by the two-step backward differentiation formula. Error estimates are proved under feasible assumptions on the regularity of the exact solution avoiding hardly fulfillable compatibility conditions. Whereas the time-weighted velocity error is of optimal second order, the time-weighted error in the pressure is of first order. Suboptimal estimates are shown for a linearisation. The results cover both the two- and three-dimensional case....

The incompressible Navier-Stokes problem is discretized in time by the two-step backward differentiation formula. Error estimates are proved under feasible assumptions on the regularity of the exact solution avoiding hardly fulfillable compatibility conditions. Whereas the time-weighted velocity error is of optimal second order, the time-weighted error in the pressure is of first order. Suboptimal estimates are shown for a linearisation. The results cover both the two- and three-dimensional...