Global Existence Of Strong Solutions Research Articles

Global strong solution to the 2D nonhomogeneous incompressible MHD system

In this paper, we first prove the unique global strong solution with vacuum to the two-dimensional nonhomogeneous incompressible MHD system, as long as the initial data satisfies some compatibility condition. As a corollary, the global existence of strong solution with vacuum to the 2D nonhomogeneous incompressible Navier–Stokes equations is also established. Our main result improves all the previous results where the initial density need to be strictly positive. The key idea is to use some critical Sobolev inequality of logarithmic type, which is originally due to Brezis and Wainger (1980) [7].

On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains

We prove global existence of strong solutions for the Vlasov-Poisson system in a convex bounded domain in the plasma physics case assuming homogeneous Dirichlet boundary conditions for the electric potential and the specular reflection boundary conditions for the distribution density.

Global existence of strong solutions of Navier-Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

The aims of this paper are to discuss global existence and uniqueness of strong solution for a class of isentropic compressible navier-Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of isentropic compressible Navier-Stokes equations. The first result shows only the existence. And the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.

A simple approach to the Cauchy problem for complex Ginzburg–Landau equations by compactness methods

This paper is concerned with the Cauchy problem (CGL) in L2(RN) for complex Ginzburg–Landau equations with Laplacian Δ and nonlinear term |u|q−2u multiplied by the complex coefficients λ+iα and κ+iβ, respectively (q⩾2, λ>0, κ>0, α,β∈R). The global existence of strong solutions to (CGL) is established without any upper restriction on q⩾2 but with some restriction on α/λ and β/κ. The result corresponds to Ginibre and Velo (1996) [3, Proposition 5.1] which is technically proved by combining convolution (regularizing) methods with compactness (localizing) methods, while our proof here is fairly simplified. The key to our proof is the Cauchy problem (CGL)R which is (CGL) with Δ replaced with Δ−VR, where VR(x):=(|x|−R)2 (|x|>R), VR(x):=0 (|x|⩽R). The solvability of (CGL)R is a direct consequence of Okazawa and Yokota (2002) [16, Theorem 4.1]. Taking the limit of global strong solutions to (CGL)R as R→∞ yields a global strong solution to (CGL). The result gives also an unbounded version of Okazawa and Yokota (2002) [16, Theorem 1.1 with p=2] for the initial–boundary value problem on bounded domains.

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

We show the existence of global strong solutions for the compressible Navier–Stokes system in dimension N⩾2 with large initial data on the rotational part of the velocity. By following Chemin and Gallagher (2009, 2011) [3,4], we aim at exhibiting large initial data u0 such that the projection on the divergence field Pu0 is large in B∞,∞−1 (which is the largest space invariant by the scaling of the equations) and such that these initial data generate global strong solution. The fact that the smallness hypothesis in Chemin and Gallagher (2009) [3] holds on the nonlinear term of convection enables us to split the solution of the compressible Navier–Stokes equations in the sum of an incompressible solution and of a purely compressible solution. Combining the notion of quasi-solution introduced in Haspot [8,9,7], we obtain the existence of global strong solution for the shallow water system for large initial velocity both on the irrotational and rotational part.

Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

We consider strong solutions to the initial boundary value problems for the isentropic compressible Navier–Stokes equations in one dimension: $$\rho\left\{\begin{array}{lll} t+(\rho u)_x=0\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\, {\rm in}\,(0,T)\times(0,1)\\ (\rho u )_t+(\rho u^2)_x+\rho \Phi_x-(\mu( \rho )u_x)_x+P_x=0\quad\quad {\rm in}\,(0,T)\times(0,1) \\\left(\left(\frac{\delta(\Phi_x)^2\,+\,1}{(\Phi_x)^2\,+\,\delta}\right)^{\frac{2-p}{2}}\Phi_x\right)_x=4\pi g(\rho-\frac{1}{|\Omega|}\int\nolimits_\Omega \rho dx\,\,\,\, )\quad\,\, {\rm in}\,(0,T)\times(0,1)\end{array}\right.$$ Here, the Φ is a non-Newtonian potential and strong solutions of the problem and obtains the uniqueness under the compatibility condition.

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

We show the existence of global strong solutions for the compressible Navier–Stokes system in dimension N⩾2 with large initial data on the irrotational part of the velocity. We introduce a new notion of quasi-solutions when the initial velocity is assumed to be irrotational, these last one exhibit regularizing effects both on the velocity and in a very surprising way also on the density (indeed the density is a priori governed by a hyperbolic equation). We would like to point out that this smoothing effect is purely non-linear and is absolutely crucial in order to deal with the pressure term as it provides new damping effects in high frequencies. In particular this new damping effect enables us to deal with a Van der Waals pressure.

GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A LONG PERIODIC DOMAIN

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a long periodic domain. We show by a simple argument that a strong solution exists globally in time when the initial velocity in <TEX>$H^1$</TEX> and the forcing function in <TEX>$L^p$</TEX>([0; T);<TEX>$L^2$</TEX>), T > 0, <TEX>$2{\leq}p{\leq}+\infty$</TEX> satisfy a certain condition. This condition common appears for the global existence in thin non-periodic domains. Larger and larger initial data and forcing functions satisfy this condition as the thickness of the domain <TEX>$\epsilon$</TEX> tends to zero.

Global weak solutions for a periodic two-component Hunter-Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component Hunter-Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then we show that the limit of approximate solutions is a global weak solution of the two-component Hunter-Saxton system.

Global Existence of Strong Solution for Equations Related to the Incompressible Viscoelastic Fluids in the Critical $L^p$ Framework

We investigate the global strong solutions for a system of equations related to the incompressible viscoelastic fluids of Oldroyd--B type with the initial data close to a stable equilibrium. We obtain the existence and uniqueness of the global solution in a functional setting invariant by the scaling of the associated equations, where the initial velocity has the same critical regularity index as the incompressible Navier--Stokes equations, and one more derivative is needed for the deformation tensor. Like the classical incompressible Navier--Stokes equations, one may construct the unique global solution for a class of large highly oscillating initial velocity. Our result also implies that the deformation tensor $F$ has the same regularity as the density of the compressible Navier--Stokes equations.

Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

We consider the global existence of strong solution u, corresponding to a class of fully nonlinear wave equations with strongly damped terms utt − kΔut = f(x, Δu) + g(x, u, Du, D2u) in a bounded and smooth domain Ω in Rn, where f(x, Δu) is a given monotone in Δu nonlinearity satisfying some dissipativity and growth restrictions and g(x, u, Du, D2u) is in a sense subordinated to f(x, Δu). By using spatial sequence techniques, the Galerkin approximation method, and some monotonicity arguments, we obtained the global existence of a solution .

Global existence and blow-up phenomena for the weakly dissipative Novikov equation

In this paper, we consider the global existence and blow-up for the weakly dissipative Novikov equation. We firstly establish the local well-posedness for the weakly dissipative Novikov equation by Kato’s theorem. Then we present some blow-up results. Finally, we present the global existence of strong solutions to the weakly dissipative equation.

Global weak solutions for a periodic two-component μ-Hunter–Saxton system

This paper is concerned with global existence of weak solutions for a periodic two-component μ-Hunter–Saxton system. We first derive global existence for strong solutions to the system with smooth approximate initial data. Then, we show that the limit of approximate solutions is a global weak solution of the two-component μ-Hunter–Saxton system.

Global existence of strong solutions to the micro-polar, compressible flow with density-dependent viscosities

This article is concerned with global strong solutions of the micro-polar, compressible flow with density-dependent viscosity coefficients in one-dimensional bounded intervals. The important point in this article is that the initial density may vanish in an open subset.

Coexistence of a diffusive predator–prey model with Holling type-II functional response and density dependent mortality

In this paper, we consider a two competitor–one prey diffusive model in which both competitors exhibit Holling type-II functional response and one of the competitors exhibits density dependent mortality rate. First, we study the local and global existence of strong solution by using the C 0 analytic semigroup. Then, we consider the local and global stability of the positive constant equilibrium by using the linearization method and Laypunov functional method, respectively. Furthermore, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain species is small or large. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system.

Global existence of strong solutions of Navier–Stokes equations with non-Newtonian potential for one-dimensional isentropic compressible fluids

The aim of this paper is to discuss the global existence and uniqueness of strong solution for a class of the isentropic compressible Navier–Stokes equations with non-Newtonian in one-dimensional bounded intervals. We prove two global existence results on strong solutions of the isentropic compressible Navier–Stokes equations. The first result shows only the existence, and the second one shows the existence and uniqueness result based on the first result, but the uniqueness requires some compatibility condition.

GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A THIN PERIODIC DOMAIN

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a thin periodic domain. We present a simple proof that a strong solution exists globally in time when the initial velocity in H¹ and the forcing function in L P (0, ∞; L²), 2 ≤ p ≤ ∞ satisfy certain condition. This condition is basically similar to that by Iftimie and Raugel[7], which covers larger and larger initial data and forcing functions as the thickness of the domain ? tends to zero.

Existence of Global Strong Solutions in Critical Spaces for Barotropic Viscous Fluids

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension N ≧ 2. We address the question of the global existence of strong solutions for initial data close to a constant state having critical Besov regularity. First, this article shows the recent results of Charve and Danchin (Arch Ration Mech Anal 198(1):233–271, 2010) and Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) with a new proof. Our result relies on a new a priori estimate for the velocity that we derive via the intermediary of the effective velocity, which allows us to cancel out the coupling between the density and the velocity as in Haspot (Well-posedness in critical spaces for barotropic viscous fluids, 2009). Second, we improve the results of Charve and Danchin (2010) and Chen et al. (2010) by adding as in Charve and Danchin (2010) some regularity on the initial data in low frequencies. In this case we obtain global strong solutions for a class of large initial data which rely on the results of Hoff (Arch Rational Mech Anal 139:303–354, 1997), Hoff (Commun Pure Appl Math 55(11):1365–1407, 2002), and Hoff (J Math Fluid Mech 7(3):315–338, 2005) and those of Charve and Danchin (2010) and Chen et al. (2010). We conclude by generalizing these results for general viscosity coefficients.

Length-scale estimates for the 3D simplified Bardina magnetohydrodynamic model

We consider the MHD-α model known as double viscous simplified Bardina MHD (SBMHD) on a 3D periodic box (MHD stands for magnetohydrodynamic). This system is a large eddy simulation model useful to approximate the turbulent behavior of an incompressible homogeneous magnetofluid because of the actual impossibility to handle the MHD model neither analytically nor via direct numerical simulation. In a previous paper (joint work with Secchi), the global existence of strong solutions to the SBMHD has been proved as well as the existence of a global attractor of finite fractal dimension. Upper bounds for such dimension are provided both in terms of the modified Grashof number and the dissipation length associated to the mean rate of energy dissipation. In this paper, we commute the same bound in an estimate in terms of the modified Reynolds number R. This result is useful because the classical Kolmogorov theory of turbulence is expressed using the Reynolds number. The global attractor estimate in terms of R is consistent with scaling theories of turbulence and can be interpreted in terms of degrees of freedom of the flow in Landau's sense. However, it does not take into account the effect of strong dissipation-range intermittency, where significant energy lies in wavenumbers larger than the inverse Kolmogorov length. Hence, we prove that there exists a series of time-averaged inverse square length-scales whose members are estimated in terms of R.

Global strong solutions of Navier–Stokes equations with interface boundary in three-dimensional thin domains

In this article, we study the spectrum of the Stokes operator in a 3D two layer domain with interface, obtain the asymptotic estimates on the spectrum of the Stokes operator as thickness ε goes to zero. Based on the spectral decomposition of the Stokes operator, a new average-like operator is introduced and applied to the study of Navier–Stokes equation in the two layer thin domains under interface boundary condition. We prove the global existence of strong solutions to the 3D Navier–Stokes equations when the initial data and external forces are in large sets as the thickness of the domain is small. This article is a continuation of our study on the Stokes operator under Navier friction boundary condition. Due to the viscosity distinction between the two layers, the Stokes operator displays radically different spectral structure from that under Navier friction boundary condition, then causes great difficulty to the analysis.