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Marius Paicu

Publications and source records attributed to Marius Paicu.

31 records · Page 2Linked to original sources
arXiv2012

Global solutions to 2-d inhomogeneous navier-stokes system with general velocity

In this paper, we are concerned with the global wellposedness of 2-D density-dependent incompressible Navier-Stokes equations with variable viscosity, in a critical functional frame- work which is invariant by the scaling of the equations and under a non-linear smallness condition on fluctuation of the initial density which has to be doubly exponential small compared with the size of the initial velocity. In the second part of the paper, we apply our methods combined with the techniques of R. Danchin and P. B. Mucha to prove the global existence of solutions to inhomogeneous Navier-Stokes system with piecewise constant initial density which has small jump at the interface and is away from vacuum. In particular, this latter result removes the smallness condition for the initial velocity in a corresponding theorem of R. Danchin and P. B. Mucha.

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arXiv2012

Global wellposedness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity

In this paper, we first prove the global existence of weak solutions to the d-dimensional incompressible inhomogeneous Navier-Stokes equations with initial data in critical Besov spaces, which satisfies a non-linear smallness condition. The regularity of the initial velocity is critical to the scaling of this system and is general enough to generate non-Lipschitz velocity field. Furthermore, with additional regularity assumption on the initial velocity or on the initial density, we can also prove the uniqueness of such solution. We should mention that the classical maximal regularity theorem for the heat kernel plays an essential role in this context.

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arXiv2010

Global existence and regularity for the full coupled Navier-Stokes and Q-tensor system

We study the global existence and regularity of solutions for a system describing the evolution of a nematic liquid crystal fluid. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. In our previous work, \cite{pz1}, we assumed that a certain parameter, $ξ$, is zero, which had the effect of cancelling certain terms. In the current work we do not make this assumption and study the full system, observing that the presence of these additional terms has a non-trivial effect, namely the quadruply exponential increase of the high norms. We also estimate differently certain terms already existent in the simplified system and improve the estimates in \cite{pz1}. We prove the existence of global weak solutions in dimensions two and three. In dimension two we prove the higher regularity of solutions and show that the high norms increase in time at most quadruply exponential. We also show the weak-strong uniqueness in dimension two.

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arXiv2010

Energy Dissipation and Regularity for a Coupled Navier-Stokes and Q-Tensor System

We study a complex non-newtonian fluid that models the flow of nematic liquid crystals. The fluid is described by a system that couples a forced Navier-Stokes system with a parabolic-type system. We prove the existence of global weak solutions in dimensions two and three. We show the existence of a Lyapunov functional for the smooth solutions of the coupled system and use the cancellations that allow its existence to prove higher global regularity, in dimension two. We also show the weak-strong uniqueness in dimension two.

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arXiv2009

Analyticity and Gevrey-class regularity for the second-grade fluid equations

We address the global persistence of analyticity and Gevrey-class regularity of solutions to the two and three-dimensional visco-elastic second-grade fluid equations. We obtain an explicit novel lower bound on the radius of analyticity of the solutions to the second-grade fluid equations that does not vanish as $t\to \infty$. Applications to the damped Euler equations are given.

math.AP↗
arXiv2009

Global Regularity for the Navier-Stokes equations with large, slowly varying initial data in the vertical direction

In a recent article, J.-Y. Chemin, I. Gallagher and M. Paicu obtained a class of large initial data generating a global smooth solution to the three dimensional, incompressible Navier-Stokes equations. This data varies slowly in the vertical direction (is a function on $εx_3$) and has a norm which blows up as the small parameter goes to zero. This type of initial data can be seen as the "ill prepared" case (in opposite with the "well prepared" case which was treated previously by J.-Y. Chemin and I. Gallagher). In that paper, the fluid evolves in a special domain, namely $Ω=T^2_h\times\R_v$. The choice of a periodic domain in the horizontal variable plays an important role. The aim of this article is to study the case where the fluid evolves in the full spaces $\R^3$, case where we need to overcome the difficulties coming from very low horizontal frequencies. We consider in this paper an intermediate situation between the "well prepared" case and "ill prepared" situation (the norms of the horizontal components of initial data are small but the norm of the vertical component blows up as the small parameter goes to zero). The proof uses the analytical-type estimates and the special structure of the nonlinear term of the equation.

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arXiv2008

Anisotropic Navier-Stokes equations in a bounded cylindrical domain

We study the global and local existence and uniqueness of solutions to the Navier-Stokes equations with anisotropic viscosity in a bounded cylindrical domain $Q=Ω\times (0,1)$, where $Ω$ is a star-shaped domain in $R^2$. In this paper, we consider the case of homogeneous Dirichlet boundary conditions on the lateral boundary and vanishing normal trace on the top and the bottom.

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arXiv2008

Global Existence Results for the Anisotropic Boussinesq System in Dimension Two

We study the global existence issue for the two-dimensional Boussinesq system with horizontal viscosity in only one equation. We first examine the case where the Navier-Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transportdiffusion equation with diffusion in the horizontal direction only. For the both systems and for arbitrarily large data, we construct global weak solutions `a la Leray. Next, we state global wellposedness results for more regular data. Our results strongly rely on the fact that the diffusion occurs in a direction perpendicular to the buoyancy force.

math.AP↗
arXiv2008

Remarks on the blow-up of solutions to a toy model for the Navier-Stokes equations

S. Montgomery-Smith provided a one dimensional model for the three dimensional, incompressible Navier-Stokes equations, for which he proved the blow up of solutions associated to a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this note the model is adapted to the case of two and three space dimensions, with the additional feature that the divergence free condition is preserved. It is checked that the family of initial data constructed previously by J.-Y Chemin and I. Gallagher which is arbitrarily large but yet generates a global solution to the Navier-Stokes equations in three space dimensions, actually causes blow up for the toy model -- meaning that the precise structure of the nonlinear term is crucial to understand the dynamics of large solutions to the Navier-Stokes equations.

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arXiv2008

Global regularity for some classes of large solutions to the Navier-Stokes equations

In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared'' (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared'' situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.

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arXiv2008

Uniform Local Existence for Inhomogeneous Rotating Fluid Equations

We investigate the equations of anisotropic incompressible viscous fluids in $\R^3$, rotating around an inhomogeneous vector $B(t, x_1, x_2)$. We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniformlocal existence result with respect to the Rossby number in the same functional spaces under the additional assumption that $B=B(t,x_1)$ or $B=B(t,x_2)$. We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law.

math.AP↗
arXiv2008

Global existence for the MHD system in critical spaces

In this article, we show that the magneto-hydrodynamic system (MHD) in $\R^N$ with variable density, variable viscosity and variable conductivity has a local weak solution in the Besov space $\dot B^{\frac{N}{p_1}}_{p_1,1}(\R^N)\times\dot B^{\frac{N}{p_2}-1}_{p_2,1}(\R^N) \times\dot B^{\frac{N}{p_2}-1}_{p_2,1}(\R^N)$ for all $1<p_2<+\infty$ and some $1<p_1\leq\frac{2N}{3}$ if the initial density approaches a positive constant. Moreover, this solution is unique if we impose the restrictive condition $1<p_2\leq2N$. We prove also that the constructed solution exist globally in time if the initial data are small enough. In particular, this allows us to work in the frame of Besov space with negative regularity indices and this fact is particularly important when the initial data are strong oscillating.

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