In this paper, we study the irrotational subsonic and subsonic-sonic flows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general case. Afterwards we obtain the subsonic-sonic limit solution by taking the extract subsonic solutions as the approximate sequences.
In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term, where the initial data lies in $W^{1,\infty}(\mathbb{R}^n) \cap C^1(\mathbb{R}^n)$. We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.
In this paper, it is proved that the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear viscosity, different far field states and periodic perturbations, not only exists globally in time, but also converges towards the viscous shock wave of the corresponding Riemann problem as time goes to infinity. Furthermore, the decay rate is shown. The proof is given by a technical energy method.
In this paper, the large time behavior of solutions of 1-D isentropic Navier-Stokes system is investigated. It is shown that a composite wave consisting of two viscous shock waves is stable for the Cauchy problem provided that the two waves are initially far away from each other. Moreover the strengths of two waves could be arbitrarily large.
In the paper, we establish the existence of steady boundary layer solution of Boltzmann equation with specular boundary condition in L_{x,v}^{2} ∩L_{x,v}^{∞} in half-space. The uniqueness, continuity and exponential decay of the solution are obtained, and such estimates are important to prove the Hilbert expansion of Boltzmann equation for half-space problem with specular boundary condition.
In 2003, Gasser-Hsiao-Li [JDE (2003)] showed that the solution to the bipolar hydrodynamic model for semiconductors (HD model) without doping function time-asymptotically converges to the diffusion wave of the porous media equation (PME) for the switch-off case. Motivated by the work of Huang-Wu [arXiv:2210.13157], we will confirm that the time-asymptotic expansion proposed by Geng-Huang-Jin-Wu [arXiv:2202.13385] around the diffusion wave is a better asymptotic profile for the HD model in this paper, where we mainly adopt the approximate Green function method and the energy method.
The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and another is a linear equation coupling the reactant concentration for the resultant concentration. This paper is concerned with the initial-boundary value problem for the above model. By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial initial-boundary value problem for Riemann type of initial-boundary data. Moreover, as examples, we apply the obtained results to the cases of head-on and wide pulse injections and give the expression of the global weak entropy solution.
In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We will prove global existence of the smooth solution to the incompressible Navier-Stokes-Landau-Lifshitz equation with small initial data in $\mathbb{T}^2$ or $\mathbb{R}^2$ and $\mathbb{R}^3$.
We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.
We study the L^{2}-supercritical nonlinear Schrödinger equation (NLS) with a partial confinement, which is the limit case of the cigar-shaped model in Bose-Einstein condensate (BEC). By constructing a cross constrained variational problem and establishing the invariant manifolds of the evolution flow, we show a sharp condition for global existence.