Global unique solutions for an initial-boundary value problem for the 2D dual-porosity-Navier-Stokes System with large initial data

We are interested in studying the Cauchy problem for the viscous shallow-water system in dimension N≥2, we show the existence of global strong solutions with large initial data on the irrotational part of the velocity for the scaling of the equations. More precisely our smallness assumption on the initial data is supercritical for the scaling of the equations. It allows us to give a first kind of answer to the problem of the existence of global strong solution with large initial energy data in dimension N=2. To do this, we introduce the notion of quasi-solutions which consists in solving the pressureless viscous shallow water system. We can obtain such solutions at least for irrotational data which are subject to regularizing effects both on the velocity and on the density. This smoothing effect is purely nonlinear and is crucial in order to build solution of the viscous shallow water system as perturbations of the “quasi-solutions”. Indeed the pressure term can be considered as a remainder term which becomes small in high frequencies for the scaling of the equations. To finish we prove the existence of global strong solution with large initial data when N≥2 provided that the Mach number is sufficiently large.

Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

The global solutions with large initial data for the isothermal compressible fluid models of Korteweg type has been studied by many authors in recent years. However, little is known of global large solutions to the nonisothermal compressible fluid models of Korteweg type up to now. This paper is devoted to this problem, and we are concerned with the global existence of smooth and non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional nonisothermal compressible fluid models of Korteweg type. The case when the viscosity coefficient $\mu(\rho)=\rho^\alpha$, the capillarity coefficient $\kappa(\rho)=\rho^\beta$, and the heat-conductivity coefficient $\tilde{\alpha}(\theta)=\theta^\lambda$ for some parameters $\alpha,\beta,\lambda\in \mathbb{R}$ is considered. Under some assumptions on $\alpha,\beta$ and $\lambda$, we prove the global existence and time-asymptotic behavior of large solutions around constant states. The proofs are given by the elementary energy method combined with the technique developed by Y. Kanel' \cite{Y. Kanel} and the maximum principle.

Existence of global strong solution for Korteweg system with large infinite energy initial data

In this paper, we consider the well-posedness of the solution for the Cauchy problem of the double-diffusive convection system in R3. We establish the local existence and uniqueness of the solution for the double-diffusive convection system in H1(R3) with large initial data and the global well-posedness under the assumption that the L2 norm of the initial data is small. Moreover, we also prove that there exists a global unique solution in HN(R3) for any N ≥ 2, without any other smallness condition of the initial data.

In this paper, we investigate the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in one space dimension. The global existence and uniqueness of strong solutions with large initial data are established when there is initial vacuum. Our result may be the first result about the global strong solution with large initial data and vacuum.

Global existence and decay rate of the Boussinesq–Burgers system with large initial data

Global solutions to 3D incompressible Navier–Stokes equations with some large initial data

In this paper, we study the barotropic compressible magnetohydrodynamic equations with the shear viscosity being a positive constant and the bulk one being proportional to a power of the density in a general two-dimensional (2D) bounded simply connected domain. For initial density allowed to vanish, we prove that the initial-boundary-value problem of a 2D compressible MHD system admits the global strong and weak solutions without any restrictions on the size of initial data provided the shear viscosity is a positive constant and the bulk one is $\lambda=\rho^\beta$ with $\beta>4/3$. As we known, this is the first result concerning the global existence of strong solutions to the compressible MHD system in general two-dimensional bounded domains with large initial data and vacuum.

In this paper, we prove the existence of global smooth solution for the Cauchy problem of nonlinearly damped p-system with large initial data. The analysis is based on several key a prioriestirmates. which are obtained by the tnaxirnum principle. Our results extend the corresponding results in [l0,11].

We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional nonisentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum---there exists a compression in the initial data. For the nonisentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum---there exists a strong compression in the initial data. Furthermore, we identify two new phenomena---decompression and de-rarefaction---for the nonisentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous nonisentropic solution, even though the initial data contain a weak compression, by solving a backward Goursat problem, so that the solution is smooth, except on several characteristic curves across which the solution has a weak discontinuity (i.e., only Lipschitz continuity). For the de-rarefaction phenomenon, we construct a continuous nonisentropic solution whose initial data contain isentropic rarefactions (i.e., without compression) and a locally stationary varying entropy profile, for which the solution still forms a shock wave in a finite time.

Existence of global strong solutions for the barotropic Navier–Stokes system with large initial data on the rotational part of the velocity

In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier–Stokes equations with vacuum in a bounded domain or exterior domain $\Omega$ of $\mathbb{R}^n$($n\ge2$). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1–28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91–129; S.J. Ding, H.Y. Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696–1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimension.

Existence of global strong solutions for the Saint-Venant system with large initial data on the irrotational part of the velocity

In this paper, we study a Neumann and free boundary problem for the one-dimensional viscous radiative and reactive gas. We prove that under rather general assumptions on the heat conductivity κ, for any arbitrary large smooth initial data, the problem admits a unique global classical solution. Our global existence results improve those results by Umehara and Tani [“Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas,” J. Differ. Equations 234(2), 439–463 (2007)10.1016/j.jde.2006.09.023; Umehara and Tani “Global solvability of the free-boundary problem for one-dimensional motion of a self-gravitating viscous radiative and reactive gas,” Proc. Jpn. Acad., Ser. A: Math. Sci. 84(7), 123–128 (2008)]10.3792/pjaa.84.123 and by Qin, Hu, and Wang [“Global smooth solutions for the compressible viscous and heat-conductive gas,” Q. Appl. Math. 69(3), 509–528 (2011)].10.1090/S0033-569X-2011-01218-0 Moreover, we analyze the asymptotic behavior of the global solutions to our problem, and we prove that the global solution will converge to an equilibrium as time goes to infinity. This is the result obtained for this problem in the literature for the first time.