Narrow Results

In this paper we consider the Rational Large Eddy Simulation model recently introduced by Galdi and Layton. We briefly present this model, which (in principle) is similar to others commonly used, and we prove the existence and uniqueness of a class of strong solutions. Contrary to the gradient model, the main feature of this model is that it allows a better control of the kinetic energy. Consequently, to prove existence of strong solutions, we do not need subgrid-scale regularization operators, as proposed by Smagorinsky. We also introduce some breakdown criteria that are related to the Euler and Navier–Stokes equations.

Rate-independent systems arise in a number of applications. Usually, weak solutions to such problems with potentially very low regularity are considered, requiring mathematical techniques capable of handling nonsmooth functions. In this work, we prove the existence of Hölder-regular strong solutions for a class of rate-independent systems. We also establish additional higher regularity results that guarantee the uniqueness of strong solutions. The proof proceeds via a time-discrete Rothe approximation and careful elliptic regularity estimates depending in a quantitative way on the (local) convexity of the potential featuring in the model. In the second part of the paper, we show that our strong solutions may be approximated by a fully discrete numerical scheme based on a spatial finite element discretization, whose rate of convergence is consistent with the regularity of strong solutions whose existence and uniqueness are established.

In this paper we obtain existence and uniqueness of solutions for the Cauchy problem for the minimizing total variation flow when the initial condition is a Radon measure in ℝ^N. We study limit solutions obtained by weakly approximating the initial measure μ by functions in L¹(ℝ^N). We are able to characterize limit solutions when the initial condition μ=h+μ_s, where h∈L¹(ℝ^N)∩L^∞(ℝ^N), and μ_s=αℋ^k⌊ S,α≥0,k is an integer and S is a k-dimensional manifold with bounded curvatures. In case k<N-1 we prove that the singular part of the solution does not move, it remains equal to μ_s for all t≥0. In particular, u(t)=δ₀ when u(0)=δ₀. In case k=N-1 we prove that the singular part of the limit solution is and we also characterize its absolutely continuous part. This explicit behaviour permits to characterize limit solutions. We also give an entropy condition characterization of the solution which is more satisfactory when k<N-1. Finally, we describe some distributional solutions which do not have the behaviour characteristic of limit solutions.

We consider a Hilbert-valued linear stochastic differential equation driven by fractional Gaussian noise. We give criteria for weak and strong solutions to exist in terms of the differential operator, which we assume to be skew-symmetric, and the covariance operator of the noise. We consider a class of PDE as examples. We show that the energy of these systems is finite if and only if a weak solution to the associated ODE exists. We also relate these conditions to conditions obtained by previous authors.

We establish the existence, uniqueness and approximation of the strong solutions for the stochastic 3D LANS-α model driven by a non-Gaussian Lévy noise. Moreover, we also study the stability of solutions. In particular, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution.

We study, in this paper, a stochastic version of a coupled Allen–Cahn–Navier–Stokes model in a two-dimensional (2D) bounded domain. The model consists of the Navier–Stokes equations (NSEs) for the velocity, coupled with a Allen–Cahn model for the order (phase) parameter. We prove the existence and the uniqueness of a variational solution.

In this paper, we derive a large deviation principle for a stochastic 2D Allen–Cahn–Navier–Stokes system with a multiplicative noise of Lévy type. The model consists of the Navier–Stokes equations for the velocity, coupled with a Allen–Cahn system for the order (phase) parameter. The proof is based on the weak convergence method introduced in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincar Probab. Stat. 47(3) (2011) 725747].

We use the approach of Röckner–Zhao to establish strong well-posedness of stochastic differential equations with singular drift satisfying some minimal assumptions.

In this paper, we consider a stochastic 2D liquid crystal system with a small multiplicative noise of Gaussian type, which models the dynamic of nematic liquid crystals under the influence of stochastic external forces. We derive a large deviation principle for the model. The proof relies on the weak convergence method that was introduced in [A. Budhiraja, P. Dupuis and V. Maroulas, Variational representations for continuous time processes, Ann. Inst. Henri Poincaré Probab. Stat. 47(3) (2011) 725–747] and based on a variational representation on infinite-dimensional Brownian motion.

We study the Cauchy problem of nonhomogeneous magneto-micropolar fluid system with zero density at infinity in the entire space ${ℝ}^2$. We prove that the system admits a unique local strong solution provided the initial density and the initial magnetic field decay not too slowly at infinity. In particular, there is no need to require any Choe–Kim type compatibility condition for the initial data.

We study the Navier–Stokes equations of three-dimensional compressible isentropic and two-dimensional heat-conducting flows in a domain Ω with nonnegative density, which may vanish in an open subset (vacuum) of Ω, and with positive density, respectively. We prove some blow-up criteria for the local strong solutions.

The three-dimensional compressible magnetohydrodynamic isentropic flow with zero magnetic diffusivity is studied in this paper. The vanishing magnetic diffusivity causes significant difficulties due to the loss of dissipation of the magnetic field. The existence and uniqueness of local-in-time strong solutions with large initial data is established. Strong solutions have weaker regularity than classical solutions. A generalized Lax–Milgram theorem and a Schauder–Tychonoff-type fixed-point argument are applied on conjunction with novel techniques and estimates for strong solutions.

In this paper, we deal with the existence of local strong solution for a perfect compressible viscous fluid, heat conductive and self-gravitating, coupled with a first-order kinetics used in astrophysical hydrodynamical models. In our setting, the vacuum is allowed and as a byproduct of the existence result we get a blow-up criterion for the local strong solution. Moreover we prove a blow-up criterion for the local strong solutions in terms of the velocity gradient, the mass fraction gradient and the temperature similar to the well-known Beale–Kato–Majda criterion for ideal incompressible flows.

In this paper we present computations of the nonweak solutions of class C¹¹ of the strong form of transient Navier–Stokes equations for compressible flow in Lagrangian frame of reference using space–time least squares finite element formulation (STLSFEF) with primitive variables ρ, u, T. For high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations (GDEs). It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations without any assumptions or approximations are possible. In the approach presented here SUPG, SUPG/DC, or SUPG/DC/LS operators are neither used nor needed. The role of diffusion, i.e., viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Merits of using ρ, u, T as primitive variables over ρ, u, p are discussed. Compression of air in a rigid cylinder by a rigid, massless and frictionless piston is used as a model problem. True time evolutions of class C¹¹ are reported beginning with the first time step until steady shock conditions are achieved. Comparisons with analytical solutions are presented when possible.

This paper presents a space–time least squares finite element formulation of one dimensional transient Navier–Stokes equations (governing differential equations: GDE) for compressible flow in Eulerian frame of reference using ρ, u, T as primitive variables with C¹¹ type p-version hierarchical interpolations in space and time. Time marching procedure is utilized to compute time evolutions for all values of time. For high speed gas dynamics the C¹¹ type interpolations in space and time possess the same orders of continuity in space and time as the GDE. It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations are possible without any assumptions or approximations. In the approach presented here SUPG, SUPG/DC, SUPG/DC/LS operators are neither used nor needed. Time evolution shows resolution of shock structure (i.e., shock speed, shock relations and shock width) to be in excellent agreement with the analytical solutions. The role of diffusion, i.e., viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Riemann shock tube is used as model problem. Time evolutions are reported beginning with the first time step until steady shock conditions are achieved. In this approach, when the computed error functionals become zero (computationally), the computed nonweak solutions satisfy GDE accurately.

The aim of this paper is to present some multiplicity results for elliptic boundary value problems involving oscillating nonlinearities. These results are obtained by using a variational theorem of Ricceri and some developments of this latter.