Local Existence Of Smooth Solutions Research Articles

Global well-posedness for a two-dimensional Keller-Segel-Euler system of consumption type

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in R2. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the W1,q-energy estimate (q>2) motivated by the partially inviscid two-dimensional Boussinesq system. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations.

Double shock solution for three-dimensional irrotational isentropic flow

We study the generalized Riemann problem for a second-order conservation law derived from the three-dimensional irrotational isentropic flow. The existence of local smooth solutions containing two shock fronts is established under minimal compatibility conditions on the initial data.

Local Existence of Smooth Solutions for the Semigeostrophic Equations on Curved Domains

Local Existence of Smooth Solutions for the Semigeostrophic Equations on Curved Domains

Global Small Solutions to the Inviscid Hall-MHD System

The local existence of smooth solutions to the inviscid Hall-MHD system has been obtained since Chae, Degond and Liu [Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, {31} (2014), 555--565]. However, as we known, how to construct the global small solutions to the inviscid Hall-MHD system is still an open problem. In the present paper, we give a positive answer in $ \mathbb{T}^3$ when the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.

Well‐posedness of initial‐boundary value problem for a coupled Benjamin–Bona–Mahony–Schrödinger system

We derive the well‐posedness results concerning the initial‐boundary value problem for a class of coupled Benjamin–Bona–Mahony (BBM)–Schrödinger system. More precisely, we obtain the existence of local smooth solutions of (1) for initial and boundary data in the energy space Hm × Hm for . Depending on a priori estimates, we also prove that the solutions exist global in time in the energy space H2 × H2.

Global Existence of Smooth Solutions for a Nonconservative Bitemperature Euler Model

The bitemperature Euler model describes a crucial step of inertial confinement fusion (ICF) when the plasma is quasineutral while ionic and electronic temperatures remain distinct. The model is written as a first-order hyperbolic system in nonconservative form with partially dissipative source terms. We consider the polytropic case for both ions and electrons with different $\gamma$-law pressures. The system does not fulfill the Shizuta--Kawashima condition and, the physical entropy, which is a strictly convex function, does not provide a symmetrizer of the system. In this paper we exhibit a symmetrizer to apply the result on the local existence of smooth solutions in several space dimensions. In the one-dimensional case we establish energy and dissipation estimates leading to global existence for small perturbations of equilibrium states.

Existence and Uniqueness of the Local Smooth Solution to 3D Stochastic MHD Equations without Diffusion.

In this paper, we consider the existence of local smooth solution to stochastic magneto-hydrodynamic equations without diffusion forced by additive noise in . We first transform the system into a random system via a simple change of variable and borrow the result obtained for classical magneto-hydrodynamic equations, then we show that this random transformed system is measurable with respect to the stochastic element. Finally we extend the solution to the maximality solution. Due to the coupled construction of this system, we need more elaborate and complicated estimates with respect to stochastic Euler equation.

Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of change variables and using the idea of characteristic decomposition, the Euler system is transformed into a new system which displays a transparent singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Finally, we obtain a local classical solution for the pseudo-steady full Euler equations by converting the solution from the partial hodograph variables to the original variables.

Local existence and blow-up criterion of 3D ideal magnetohydrodynamics equations

We investigate the local existence of smooth solutions of a 3D ideal magnetohydrodynamics (MHD) equations in a bounded domain and give a blow-up criteria to this equations with respect to vorticists.

Local existence and singularities formation for isothermal relativistic radiation hydrodynamics equations

We study the Cauchy problem for multi-dimensional compressible relativistic hydrodynamics in the presence of a radiation field. First, based on the theory of quasilinear symmetric hyperbolic, we establish the local existence of smooth solutions for both non-vacuum and vacuum cases. Next, in the spirit of Sideris’ work [T. Sideris, Formation of singularities of solutions to nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 86 (1984) 369–381; T. Sideris, Formation of singularities in three-dimensional compressible fluids, Comm. Math. Phys. 101 (1985) 475–485], we show that smooth solutions blow-up in finite time if the initial data is compactly supported and large enough. Compared with the previous work, the main difficulties of the first problem lie in two aspects, we must first deal with the source terms relying on radiative quantities, and we also need to solve out the new coefficients matrices under the Lorentz transformation for vacuum case. The second difficulty arises on how to verifying that the smooth solution has finite propagation speed..

Probabilistic representation and local existence for the quasi-linear partial integro-differential equations with Sobolev initial value

In this paper, we establish the stochastic representation for quasi-linear partial integro-differential equations. Further, by using it, we prove the existence of local smooth solutions in Sobolev spaces for a class of quasi-linear partial integro-differential equations with smooth coefficients.

The Cauchy problem for a Bardina–Oldroyd model to the incompressible viscoelastic flow

In this paper, we study the Cauchy problem for a regularized viscoelastic fluid model in space dimension two, the Bardina–Oldroyd model, which is inspired by the simplified Bardina model for the turbulent flows of fluids, introduced by Cao et al. (2006). In particular, we obtain the local existence of smooth solutions to this model via the contraction mapping principle. Furthermore, we prove the global existence of smooth solutions to this system.

Smooth solutions for one‐dimensional relativistic radiation hydrodynamic equations

In this research article, we investigated the existence of local smooth solutions for relativistic radiation hydrodynamic equations in one spatial variable. The proof is based on a classical iteration method and the Banach contraction mapping principle. However, because of the complexity of relativistic radiation hydrodynamics equations, we first rewrite this system into a semilinear form to construct the iteration scheme and then use left eigenvectors to decouple the system instead of applying standard energy method on symmetric hyperbolic systems. Different from multidimensional case, we just use the characteristic method, which can keep the properties of the initial data. Copyright © 2015 John Wiley & Sons, Ltd.

Well-posedness for Hall-magnetohydrodynamics

We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resistive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions.

Approximation of a compressible Euler–Poisson equations by a non-isentropic Euler–Maxwell equations

In this article, the convergence of time-dependent and non-isentropic Euler–Maxwell equations to compressible Euler–Poisson equations in a torus via the non-relativistic limit is studied. The local existence of smooth solutions to both equations is proved by using energy method for first order symmetrizable hyperbolic systems. The method of asymptotic expansion and the symmetric hyperbolic property of the systems are used to justify the convergence of the limit.

Incompressible type euler as scaling limit of compressible Euler-Maxwell equations

In this paper, we study the convergence of time-dependent Euler-Maxwell equations to incompressible type Euler equations in a torus via the combined quasi-neutral and non-relativistic limit. For well prepared initial data, the local existence of smooth solutions to the limit equations is proved by an iterative scheme. Moreover, the convergences of solutions of the former to the solutions of the latter are justified rigorously by an analysis of asymptotic expansions and the symmetric hyperbolic property of the systems.

Local existence of smooth solutions to the FENE dumbbell model

The author proves the local existence of smooth solutions to the finite extensible nonlinear elasticity (FENE) dumbbell model of polymeric flows in some weighted spaces if the non-dimensional parameter b > 2.

Probabilistic approach for systems of second order quasi-linear parabolic PDEs

Using the stochastic representation for second order parabolic equations, we prove the existence of local smooth solutions in Sobolev spaces for a class of second order quasi-linear parabolic partial differential equations (possibly degenerate) with smooth coefficients. As a simple application, the rate of convergence for vanishing viscosity is proved to be O(νt). Moreover, using Bismutʼs formula, we also obtain a global existence result for non-degenerate semi-linear parabolic equations. In particular, multi-dimensional Burgers equations are covered.

Vanishing viscosity limit for a coupled Navier–Stokes/Allen–Cahn system

In this paper, we study the vanishing viscosity limit for a coupled Navier–Stokes/Allen–Cahn system in a bounded domain. We first show the local existence of smooth solutions of the Euler/Allen–Cahn equations by modified Galerkin method. Then using the boundary layer function to deal with the mismatch of the boundary conditions between Navier–Stokes and Euler equations, and assuming that the energy dissipation for Navier–Stokes equation in the boundary layer goes to zero as the viscosity tends to zero, we prove that the solutions of the Navier–Stokes/Allen–Cahn system converge to that of the Euler/Allen–Cahn system in a proper small time interval. In addition, for strong solutions of the Navier–Stokes/Allen–Cahn system in 2D, the convergence rate is c ν 1 / 2 .

Smooth solutions of non-linear stochastic partial differential equations driven by multiplicative noises

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.