Regularity and attractors for the three‐dimensional generalized Boussinesq system

A Boltzmann equation, used to describe the evolution of the density function of a gas of photons interacting by Compton scattering with electrons at low density and non relativistic equilibrium, is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions is proved for a large set of initial data. A simplified equation, where only the quadratic terms are kept and that appears at very low temperature of the electron gas, for small values of the photon's energies, is also studied. The existence of weak solutions, and also of more regular solutions that are very flat near the origin, is proved. The long time asymptotic behavior of weak solutions of the simplified equation is described.

This paper deals with the one-dimensional nonlinear thermoviscoelastic system subject to constant temperature boundary conditions, describing phase transitions in shape memory alloys. We shall prove the global existence and uniqueness of the weak solution for initial data (strain, velocity, absolute temperature) (u0, v0, θ0)∈L∞×W01,∞×H1. Furthermore, we investigate the asymptotic behavior as time tends to infinity and establish the following asymptotic properties for the weak solution: As t→∞, v→0 in H1(0, 1), θ→T0 in L∞(0, 1), u(x, t)→u∞(x) a.e., where u∞∈L∞ is a stationary state satisfying f1(u∞)T0+f2(u∞)=0, a.e., in [0, 1]. Here we work in a framework in which u belongs to L∞ to describe phase transitions between different configurations of crystal lattices.

In this paper, we are concerned with the Cauchy problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity μ(ρ) = ρα(α > 0) and pressure P(ρ) = ργ (γ > 1). We will establish the global existence and asymptotic behavior of weak solutions for any α > 0 and γ > 1 under the assumption that the density function keeps a constant state at far fields. In particular, in the case that \documentclass[12pt]{minimal}\begin{document}$0<\alpha <\frac{1}{2}$\end{document}0<α<12, we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).

In this paper, we are concerned with the asymptotic behaviour of weak solutions to the initial boundary value problem for a class of quasilinear hyperbolic equations with damping terms involving p-Laplacian. By using the multiplier methods, we investigate the stability of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on strain-caused stress term and damping terms.

We investigate the asymptotic behavior of weak solutions to the semilinear non autonomous wave equation¶¶ $ u_{tt} - \Delta u + u_t |u_t|^{p-1} = V(t)u |u |^{p-1} + f(.,t), $ ¶¶ where V (t) is a positive time dependent potential satisfying¶¶ $ V(t) = O((1 + t)^{- \lambda})\,\, {\rm as} \, t \rightarrow + \infty $ ¶¶ and f t decays to 0 as $ t \rightarrow + \infty $ . We show that for $ 0 \le \lambda \le p $ there are initial values such that the energy norm of the corresponding solutions grows at least polynomially as $ t \rightarrow + \infty $ , while if $ \lambda \ge p $ the energy norm remains uniformly bounded for any choice of initial values; moreover, in certain cases there is an absorbing ball for the orbits.

In this paper, the Cauchy problem for nonstrictly hyperbolic systems of conservation laws modeling polymer flooding in enhanced oil recovery is studied, and a short proof of global existence of weak entropy solutions is obtained by using a new technique from the compensated compactness theorem coupled with the total variation estimate on one family of Riemann invariants. This work extends in some sense the previous works [T. Barkve, SIAM J. Appl. Math., 49 (1989), pp. 789--798; T. Johansen and R. Winther, SIAM J. Math. Anal., 19 (1988), pp. 541--566] about the global existence of weak solutions for the Riemann problem and by the author [J. Funct. Anal., 261 (2011), pp. 2797--2815] about the global weak solution of the Cauchy problem for an extended hyperbolic system of Keyfitz--Kranzer or Aw--Rascle type.

We study the initial-boundary value problem for 1D compressible magnetohydrodynamics equations of viscous nonresistive fluids in the Lagrangian mass coordinates. Based on the estimates of upper and lower bounds of the density, weak solutions are constructed by approximation of global regular solutions, the existence of which has recently been obtained by Jiang and Zhang [Nonlinearity 30, 3587–3612 (2017)]. The uniqueness of weak solutions is also proved as a consequence of Lipschitz continuous dependence on the initial data. Furthermore, long time behavior for global solutions is investigated. Specifically, based on the uniform-in-time bounds of the density from above and below away from zero, together with the structure of the equations, we show the exponential decay rate in L2- and H1-norm, respectively, with large initial data.

On a nonlocal Cahn-Hilliard/Navier-Stokes system with degenerate mobility and singular potential for incompressible fluids with different densities

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

Global weak solutions to a two-dimensional compressible MHD equations of viscous non-resistive fluids

Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier–Stokes system with vacuum and large initial data

1. An extensive literature is devoted to systems of quasilinear equations of motion of viscous compressible media (see [1, 2]). One of the problems of current interest is the development of the theory of global (with respect to time and data) weak solutions, including questions of their existence, uniqueness, and continuous dependence on data (such as initial data, the coefficients of the equations, the constant terms, etc.). For the system of equations of one-dimensional motion of a viscous heat-conducting gas, the last two questions were studied in [3–6] (see also [7, 8]). For more complicated systems of equations for reacting gases (in other words, in combustion problems) weak solutions were studied in [9, 10]. Quite recently, in [11], the existence of weak solutions was obtained for an even more complicated system of equations, namely, for a reacting binary gas mixture. Note that regular solutions of problems concerning binary mixtures were studied earlier in [12, 13]. The present paper is devoted to the questions of uniqueness of these weak solutions and of their continuous dependence on data. The system of quasilinear equations of one-dimensional motion of a viscous compressible reacting binary gas mixture is of the form

We study the Cauchy problem for the Schrödinger-improved Boussinesq system in a two-dimensional domain. Under natural assumptions on the data without smallness, we prove the existence and uniqueness of global strong solutions. Moreover, we consider the vanishing “improvement” limit of global solutions as the coefficient of the linear term of the highest order in the equation of ion sound waves tends to zero. Under the same smallness assumption on the data as in the Zakharov case, solutions in the vanishing “improvement” limit are shown to satisfy the Zakharov system.

We study the convergence of homoclinic orbits and heteroclinic orbits in the dynamical system governing traveling wave solutions of a perturbed Boussinesq systems modeling two-directional propagation of water waves. Nonanalytic weak solutions are found to be limits of these orbits, including compactons, peakons, and rampons, as well as infinitely many mesaons occurring at the same fixed point in the dynamical system. Singularities of solitary wave solutions in the system are also studied to understand the important impact of both linear and nonlinear dispersion terms on the regularity of these solutions.

Consider a smooth bounded domain \Omega\subseteq\mathbb R^3 with boundary \partial\Omega , a time interval [0,T) , with T\in(0,\infty] , and the Navier–Stokes system in [0,T) \times \Omega , with initial value u_0 \in L^2_{\sigma} (\Omega) and external force f= {\mathrm{div}}\,F , F \in L^2 (0,T;L^2(\Omega)) . Our aim is to extend the well-known class of Leray-Hopf weak solutions u satisfying u_{\vert{\partial \Omega}}=0 , {\mathrm{div}}\,u=0 to the more general class of Leray-Hopf type weak solutions u with general data u_{\vert{\partial \Omega}} =g , {\mathrm{div}}\,u=k satisfying a certain energy inequality. Our method rests on a perturbation argument writing u in the form u=v+E with some vector field E in [0,T)\times \Omega satisfying the (linear) Stokes system with f=0 and nonhomogeneous data. This reduces the general system to a perturbed Navier–Stokes system with homogeneous data, containing an additional perturbation term. Using arguments as for the usual Navier–Stokes system we get the existence of global weak solutions for the more general system.