305 research outputs found

    Global Well-posedness of the 3D Primitive Equations With Partial Vertical Turbulence Mixing Heat Diffusion

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    The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data

    Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations

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    In light of the question of finite-time blow-up vs. global well-posedness of solutions to problems involving nonlinear partial differential equations, we provide several cautionary examples which indicate that modifications to the boundary conditions or to the nonlinearity of the equations can effect whether the equations develop finite-time singularities. In particular, we aim to underscore the idea that in analytical and computational investigations of the blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary conditions may need to be taken into greater account. We also examine a perturbation of the nonlinearity by dropping the advection term in the evolution of the derivative of the solutions to the viscous Burgers equation, which leads to the development of singularities not present in the original equation, and indicates that there is a regularizing mechanism in part of the nonlinearity. This simple analytical example corroborates recent computational observations in the singularity formation of fluid equations

    A Blow-Up Criterion for the 3D Euler Equations Via the Euler-Voigt Inviscid Regularization

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    We propose a new blow-up criterion for the 3D Euler equations of incompressible fluid flows, based on the 3D Euler-Voigt inviscid regularization. This criterion is similar in character to a criterion proposed in a previous work by the authors, but it is stronger, and better adapted for computational tests. The 3D Euler-Voigt equations enjoy global well-posedness, and moreover are more tractable to simulate than the 3D Euler equations. A major advantage of these new criteria is that one only needs to simulate the 3D Euler-Voigt, and not the 3D Euler equations, to test the blow-up criteria, for the 3D Euler equations, computationally

    Persistency of Analyticity for Nonlinear Wave Equations: An Energy-like Approach

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    We study the persistence of the Gevrey class regularity of solutions to nonlinear wave equations with real analytic nonlinearity. Specifically, it is proven that the solution remains in a Gevrey class, with respect to some of its spatial variables, during its whole life-span, provided the initial data is from the same Gevrey class with respect to these spatial variables. In addition, for the special Gevrey class of analytic functions, we find a lower bound for the radius of the spatial analyticity of the solution that might shrink either algebraically or exponentially, in time, depending on the structure of the nonlinearity. The standard L2L^2 theory for the Gevrey class regularity is employed; we also employ energy-like methods for a generalized version of Gevrey classes based on the ℓ1\ell^1 norm of Fourier transforms (Wiener algebra). After careful comparisons, we observe an indication that the ℓ1\ell^1 approach provides a better lower bound for the radius of analyticity of the solutions than the L2L^2 approach. We present our results in the case of period boundary conditions, however, by employing exactly the same tools and proofs one can obtain similar results for the nonlinear wave equations and the nonlinear Schr\"odinger equation, with real analytic nonlinearity, in certain domains and manifolds without physical boundaries, such as the whole space Rn\mathbb{R}^n, or on the sphere Sn−1\mathbb{S}^{n-1}

    Onsager's Conjecture for the Incompressible Euler Equations in Bounded Domains

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    The goal of this note is to show that, also in a bounded domain Ω⊂Rn\Omega \subset \mathbb{R}^n, with ∂Ω∈C2\partial \Omega\in C^2, any weak solution, (u(x,t),p(x,t))(u(x,t),p(x,t)), of the Euler equations of ideal incompressible fluid in Ω×(0,T)⊂Rn×Rt\Omega\times (0,T) \subset \mathbb{R}^n\times\mathbb{R}_t, with the impermeability boundary condition: u⋅n⃗=0u\cdot \vec n =0 on ∂Ω×(0,T)\partial\Omega\times(0,T), is of constant energy on the interval (0,T)(0,T) provided the velocity field u∈L3((0,T);C0,α(Ω‾))u \in L^3((0,T); C^{0,\alpha}(\overline{\Omega})), with $\alpha>\frac13\,.

    On the Global Regularity of a Helical-decimated Version of the 3D Navier-Stokes Equations

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    We study the global regularity, for all time and all initial data in H1/2H^{1/2}, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution of Navier-Stokes (NS) equations into the subspace where helicity (the L2−L^2-scalar product of velocity and vorticity) is sign-definite. The presence of a second (beside energy) sign-definite inviscid conserved quadratic quantity, which is equivalent to the H1/2−H^{1/2}-Sobolev norm, allows us to demonstrate global existence and uniqueness, of space-periodic solutions, together with continuity with respect to the initial conditions, for this decimated 3D model. This is achieved thanks to the establishment of two new estimates, for this 3D model, which show that the H1/2H^{1/2} and the time average of the square of the H3/2H^{3/2} norms of the velocity field remain finite. Such two additional bounds are known, in the spirit of the work of H. Fujita and T. Kato \cite{kato1,kato2}, to be sufficient for showing well-posedness for the 3D NS equations. Furthermore, they are directly linked to the helicity evolution for the dNS model, and therefore with a clear physical meaning and consequences
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