Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Global well posedness for the Maxwell–Navier–Stokes system in 2D

We prove global existence of regular solutions to the full MHD system (or more precisely the Maxwell–Navier–Stokes system) in 2D. We also provide an exponential growth estimate for the Hs norm of the solution when the time goes to infinity.     

On a lemma due to Ladyzhenskaya and Solonnikov and some applications

We present some applications of a lemma by Ladyzhenskaya and Solonnikov [Determination of solutions of boundary value problems for stationary Stokes and Navier–Stokes equations having an unbounded Dirichlet integral, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980) 117–160 (English Transl.: J. Soviet Math. 21 (1983) 728–761)]. Some other results in that paper referring to stationary Navier–Stokes equations are extended to a non-Newtonian fluid, the so-called micropolar fluid. This model depends on the microrotational viscosity _νr${\nu }_{\mathrm{r}}$ which vanishes for a Navier–Stokes fluid. We use the lemma in full to show that, as _νr${\nu }_{\mathrm{r}}$ tends to zero, the solutions of the Ladyzhenskaya–Solonnikov problem converge to the solutions of the corresponding problem for Navier–Stokes equations. In addition, we obtain a similar convergence regarding the Leray problem for micropolar fluids.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic–parabolic systems

Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the Navier–Stokes equations of compressible gas, and magnetohydrodynamics with inflow or outflow boundary conditions, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index γ?1. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive (“shock-type”) layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.     

Some properties of a class of abstract stationary equations

We study a class of abstract nonlinear equations in a separable Hilbert space for which we prove some generic properties of the set of solutions. The results apply, in particular, in several models of hydrodynamics, such as magneto-micropolar equations, micropolar fluid equations, Boussinesq and Navier–Stokes equations.     

Unicité des solutions des équations de Navier–Stokes dans les espaces de Morrey–Campanato

In a recent work [P.G. Lemarié-Rieusset, Uniqueness for the Navier–Stokes problem: Remarks on a theorem of Jean-Yves Chemin, Nonlinearity 20 (2007) 1475–1490], P.G. Lemarié-Rieusset proved the uniqueness of solution to the Navier–Stokes equations in the space provided that p>2 and q>d. In this paper, we prove a local version of this result which covers the limit case q=d. Precisely, we prove the uniqueness of solution to the Navier–Stokes equations in the space for every p>2 and r>2 where is the closure of the test functions in the Morrey–Campanato space Mr,d(Rd). The prove of our result relies on an extension of the Comparison Theorem of P.G. Lemarié-Rieusset (Theorem 21.1 in [P.G. Lemarié-Rieusset, Recent developments in the Navier–Stokes problem, Chapman & Hall/CRC, 2002]). Moreover, this extension allows us to prove the uniqueness of solution to the Navier–Stokes equations in a functional space closed to the critical space C([0,T],M2,d(Rd)).     

Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers

The present paper seeks to continue the analysis in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on tangential boundary stabilization of Navier–Stokes equations, d=2,3$d=2,3$, as deduced from well-posedness and stability properties of the corresponding linearized equations. It intends to complement [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on two levels: (i) by casting the Riccati-based results of Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] for d=2,3$d=2,3$ in an abstract setting, thus extracting the key relevant features, so that the resulting framework may be applicable also to other stabilizing boundary feedback operators, as well as to other parabolic-like equations of fluid dynamics; (ii) by including, in the case d=2$d=2$ this time, also the low-level gain counterpart of the results in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] with both Riccati-based and spectral-based (tangential) feedback controllers. This way, new local boundary stabilization results of Navier–Stokes equations are obtained over [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear].     

Large deviations for the two-dimensional Navier–Stokes equations with multiplicative noise

A Wentzell–Freidlin type large deviation principle is established for the two-dimensional Navier–Stokes equations perturbed by a multiplicative noise in both bounded and unbounded domains. The large deviation principle is equivalent to the Laplace principle in our function space setting. Hence, the weak convergence approach is employed to obtain the Laplace principle for solutions of stochastic Navier–Stokes equations. The existence and uniqueness of a strong solution to (a) stochastic Navier–Stokes equations with a small multiplicative noise, and (b) Navier–Stokes equations with an additional Lipschitz continuous drift term are proved for unbounded domains which may be of independent interest.     

Regularity of weak solutions to 3D incompressible Navier–Stokes equations

In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with ${r \in [1, \infty)}In this paper, we establish some new local and global regularity properties for weak solutions of 3D non-stationary Navier–Stokes equations in the class of L ^r (0, T ; L ³(Ω)) with r ? [1, ￥){r \in [1, \infty)} , which are beyond Serrin’s condition.     

Martingale solutions and Markov selections for stochastic partial differential equations

We present a general framework for solving stochastic porous medium equations and stochastic Navier–Stokes equations in the sense of martingale solutions. Following Krylov [N.V. Krylov, The selection of a Markov process from a Markov system of processes, and the construction of quasidiffusion processes, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973) 691–708] and Flandoli–Romito [F. Flandoli, N. Romito, Markov selections for the 3D stochastic Navier–Stokes equations, Probab. Theory Related Fields 140 (2008) 407–458], we also study the existence of Markov selections for stochastic evolution equations in the absence of uniqueness.     

A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations.     

Periodic solutions of the Navier�CStokes equations with the inhomogeneous time-dependent boundary data under the general flux condition

We consider the initial boundary value problem to the Navier–Stokes equations in a bounded domain with the inhomogeneous time-dependent data b(t) ? H^1/2(?W){\beta(t) \in H^{1/2}(\partial\Omega)} under the general flux condition. We establish a reproductive property for weak solutions of the Navier–Stokes equations. Here, the reproductive property is regarded as the generalization of the time periodicity. As an application, we can prove the existence of periodic weak solutions.     

A blow-up criterion for compressible viscous heat-conductive flows

We study an initial boundary value problem for the three-dimensional Navier–Stokes equations of viscous heat-conductive fluids in a bounded smooth domain. We establish a blow-up criterion for the local strong solutions in terms of the temperature and the gradient of velocity only, similar to the Beale–Kato–Majda criterion for ideal incompressible flows.     

The Cauchy Problem for Second-Order Elliptic Systems on the Plane

Regular solutions to second-order elliptic systems on the plane are representable in terms of A-analytic functions satisfying an operator equation of the Beltrami type. We prove Carleman-type formulas for reconstruction of solutions from data on a part of the boundary of the domain. We use these formulas for solving the Cauchy problems for the system of Lame equations, the Navier–Stokes system, and the system of equations of elasticity with resilience.     

A regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations

A spatio-temporal localization of the BMO-version of the Beale–Kato–Majda criterion for the regularity of solutions to the 3D Navier–Stokes equations obtained by Kozono and Taniuchi, i.e., the time-integrability of the BMO-norm of the vorticity, is presented.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Application of finite element method in aeroelasticity

The subject of this paper is the numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil, which can rotate around the elastic axis and oscillate in the vertical direction. The numerical simulation consists of the finite element approximation of the Navier–Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The arbitrary Lagrangian–Eulerian (ALE) formulation of the Navier–Stokes equations, stabilization the finite element discretization and coupling of both models is discussed. Moreover, the Reynolds averaged Navier–Stokes (RANS) system of equations together with the Spallart–Almaras turbulence model is also discussed. The computational results of aeroelastic calculations are presented and compared with the NASTRAN code solutions.     

On signed measure valued solutions of stochastic evolution equations

We study existence, uniqueness and mass conservation of signed measure valued solutions of a class of stochastic evolution equations with respect to the Wiener sheet, including as particular cases the stochastic versions of the regularized two-dimensional Navier–Stokes equations in vorticity form introduced by Kotelenez.     

Higher derivatives estimate for the 3D Navier–Stokes equation

In this article, a nonlinear family of spaces, based on the energy dissipation, is introduced. This family bridges an energy space (containing weak solutions to Navier–Stokes equation) to a critical space (invariant through the canonical scaling of the Navier–Stokes equation). This family is used to get uniform estimates on higher derivatives to solutions to the 3D Navier–Stokes equations. Those estimates are uniform, up to the possible blowing-up time. The proof uses blow-up techniques. Estimates can be obtained by this means thanks to the galilean invariance of the transport part of the equation.