Leray-Hopf Solution Research Articles

Remarks on the separation of Navier–Stokes flows

Recently, strong evidence has accumulated that some solutions to the Navier–Stokes equations in physically meaningful classes are not unique. The primary purpose of this paper is to establish necessary properties for the error of hypothetical non-unique Navier–Stokes flows under conditions motivated by the scaling of the equations. Our first set of results show that some scales are necessarily active—comparable in norm to the full error—as solutions separate. ‘Scale’ is interpreted in several ways, namely via algebraic bounds, the Fourier transform and discrete volume elements. These results include a new type of uniqueness criteria which is stated in terms of the error. The second result is a conditional predictability criteria for the separation of small perturbations. An implication is that the error necessarily activates at larger scales as flows de-correlate. The last result says that the error of the hypothetical non-unique Leray–Hopf solutions of Jia and Šverák locally grows in a self-similar fashion. Consequently, within the Leray–Hopf class, energy can hypothetically de-correlate at a rate which is faster than linear. This contrasts numerical work on predictability which identifies a linear rate. Our results suggest that this discrepancy may be explained by the fact that non-uniqueness might arise from perturbation around a singular flow.

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α≥5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1−2α,∞,p) and (2α/γ+1−2α,γ,∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff Hη⁎ measure, where η⁎>0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.

Layer separation of the 3D incompressible Navier–Stokes equation in a bounded domain

We provide an unconditional L 2 upper bound for the boundary layer separation of Leray–Hopf solutions in a smooth bounded domain. By layer separation, we mean the discrepancy between a (turbulent) low-viscosity Leray–Hopf solution u ν and a fixed (laminar) regular Euler solution u ¯ with similar initial conditions and body force. We show an asymptotic upper bound C | | u ¯ | | L ∞ 3 T on the layer separation, anomalous dissipation, and the work done by friction. This extends the previous result when the Euler solution is a regular shear in a finite channel. The key estimate is to control the boundary vorticity in a way that does not degenerate in the vanishing viscosity limit.

Energy equality of the weak solutions to Navier–Stokes equations in the multiplier spaces

In this paper, we study the problem of energy conservation for the Leray–Hopf solutions of the Cauchy problem associated to the 3D incompressible Navier–Stokes equations. We identify some sufficient conditions, involving the Sobolev multiplier spaces, of the validity of the energy equality.

Non‐uniqueness of Leray–Hopf solutions to the forced fractional Navier–Stokes equations in three dimensions, up to the J. L. Lions exponent

AbstractIn this paper, we show that for , there exists a force and two distinct Leray–Hopf flows solving the forced fractional Navier–Stokes equation starting from rest. This shows that the J. L. Lions exponent is sharp in the class of Leray–Hopf solutions for the forced fractional Navier–Stokes equation.

Variational resolution of outflow boundary conditions for incompressible Navier–Stokes

This paper focuses on the so-called weighted inertia-dissipation-energy variational approach for the approximation of unsteady Leray–Hopf solutions of the incompressible Navier–Stokes system. Initiated in (Ortiz et al 2018 Nonlinearity 31 5664–82), this variational method is here extended to the case of non-Newtonian fluids with power-law index r ⩾ 11/5 in three space dimension and large nonhomogeneous data. Moreover, boundary conditions are not imposed on some parts of boundaries, representing, e.g., outflows. Correspondingly, natural boundary conditions arise from the minimisation. In particular, at walls we recover boundary conditions of Navier-slip type. At outflows and inflows, we obtain the condition . This provides the first theoretical explanation for the onset of such boundary conditions.

Leray–Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress–strain relation

We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba–Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.

Non Uniqueness of Power-Law Flows

We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension dge 3. For the power index q below the compactness threshold, i.e. q in (1, frac{2d}{d+2}), we show ill-posedness of Leray–Hopf solutions. For a wider class of indices q in (1, frac{3d+2}{d+2}) we show ill-posedness of distributional (non-Leray–Hopf) solutions, extending the seminal paper of Buckmaster & Vicol [10]. In this wider class we also construct non-unique solutions for every datum in L^2.

Nonuniqueness of Leray–Hopf solutions for a dyadic model

Nonuniqueness of Leray–Hopf solutions for a dyadic model

About local continuity with respect to $$L_{2}$$ initial data for energy solutions of the Navier–Stokes equations

In this paper we consider classes of initial data that ensure local-in-time Hadamard well-posedness of the associated weak Leray–Hopf solutions of the three-dimensional Navier–Stokes equations. In particular, for any solenodial $$L_{2}$$ initial data $$u_{0}$$ belonging to certain subsets of $$VMO^{-1}(\mathbb {R}^3)$$ , we show that weak Leray–Hopf solutions depend continuously with respect to small divergence-free $$L_{2}$$ perturbations of the initial data $$u_{0}$$ (on some finite-time interval). Our main result is inspired by and improves upon previous work of the author (Barker in J Math Fluid Mech 20(1):133–160, 2018) and work of Jean–Yves Chemin (Commun Pure Appl Math 64(12):1587–1598, 2011). Our method builds upon [4, 9]. In particular our method hinges on decomposition results for the initial data inspired by Calderon (Trans Am Math Soc 318(1):179–200, 1990) together with use of persistence of regularity results. The persistence of regularity statement presented may be of independent interest, since it does not rely upon the solution or the initial data being in the perturbative regime.

On the Shinbrot’s criteria for energy equality to Newtonian fluids: A simplified proof, and an extension of the range of application

We show that the classical Shinbrot’s criteria to guarantee that a Leray–Hopf solution satisfies the energy equality follows trivially from the L4((0,T)×Ω)) Lions–Prodi particular case. Moreover we extend Shinbrot’s result to space coefficients r∈(3,4). In this last case our condition coincides with Shinbrot condition for r=4, but for r<4 it is more restrictive than the classical one, 2∕p+2∕r=1. It looks significant that in correspondence to the extreme values r=3 and r=∞, and just for these two values, the conditions become respectively u∈L∞(L3) and u∈L2(L∞), which imply regularity by appealing to classical Ladyzhenskaya–Prodi–Serrin (L–P–S) type conditions. However, for values r∈(3,∞) the L–P–S condition does not apply, even for the more demanding case 3<r<4. The proofs are quite trivial, by appealing to interpolation, with L∞(L2) in the first case and with L2(L6) in the second case. The central position of this old classical problem in Fluid-Mechanics, together with the simplicity of the proofs (in particular the novelty of the second result) looks at least curious. This may be considered a merit of this very short note.

Partial and full hyper-viscosity for Navier-Stokes and primitive equations

The 3-D primitive equations and incompressible Navier-Stokes equations with either full hyper-viscosity or only horizontal hyper-viscosity are considered on the torus, i.e., the diffusion term −Δ is replaced by −Δ+ε(−Δ)l or by −Δ+ε(−ΔH)l, respectively, where ΔH=∂x2+∂y2, Δ=ΔH+∂z2, ε>0, l>1. Hyper-viscosity is applied in many numerical schemes, and in particular horizontal hyper-viscosity appears in meteorological models. A classical result by Lions states that for the Navier-Stokes equations uniqueness of global weak Leray-Hopf solutions for initial data in L2 holds if −Δ is replaced by (−Δ)5/4. Here, for the primitive equations the corresponding result is proven for (−Δ)8/5. For the case of horizontal hyper-viscosity l=2 is sufficient in both cases. Strong convergence for ε→0 of hyper-viscous solutions to a weak solution of the Navier-Stokes and primitive equations, respectively, is proven as well. The approach presented here is based on the construction of strong solutions via an evolution equation approach for initial data in L2 and weak-strong uniqueness.

A global well-posedness result for the Rosensweig system of ferrofluids

In this paper we study a Bloch–Torrey regularization of the Rosensweig system for ferrofluids. The scope of this paper is twofold. First of all, we investigate the existence and uniqueness of Leray–Hopf solutions of this model in the whole space \mathbb R^2 . In the second part of this paper we investigate both the long-time behavior of weak solutions and the propagation of Sobolev regularities in dimension two.

On the relation between very weak and Leray–Hopf solutions to Navier–Stokes equations

We prove a general result that implies that very weak solutions to the Cauchy problem for the Navier–Stokes equations must be, in fact, Leray–Hopf solutions if only their initial data are (solenoidal) with finite kinetic energy.

Blowup criterion for Navier–Stokes equation in critical Besov space with spatial dimensions d ≥ 4

This paper is concerned with the blowup criterion for mild solution to the incompressible Navier–Stokes equation in higher spatial dimensions d≥4. By establishing an ϵ regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in B˙p,q−1+d/p(Rd), d<p,q<∞ becomes singular at a finite time T⁎, thenlimsupt→T⁎‖u(t)‖B˙p,q−1+d/p(Rd)=∞. The corresponding result in 3D case has been obtained in [24]. As a by-product, we also prove a regularity criterion for the Leray–Hopf solution in the critical Besov space, which generalizes the results in [17], where blowup criterion in critical Lebesgue space Ld(Rd) is addressed.

Infinitely many Leray–Hopf solutions for the fractional Navier–Stokes equations

We prove the ill-posedness for the Leray–Hopf weak solutions of the incompressible and ipodissipative Navier–Stokes equations, when the power of the diffusive term is . We construct infinitely many solutions, starting from the same initial datum, which belong to and strictly dissipate their energy in small time intervals. The proof exploits the “convex integration scheme” introduced by C. De Lellis and L. Székelyhidi for the incompressible Euler equations, joining these ideas with new stability estimates for a class of non-local advection-diffusion equations and a local (in time) well-posedness result for the fractional Navier–Stokes system. Moreover, we show the existence of dissipative Hölder continuous solutions of Euler equations that can be obtained as a vanishing viscosity limit of Leray–Hopf weak solutions of suitable fractional Navier–Stokes equations.

A note on the $3$-D Navier-Stokes equations

We consider the Navier-Stokes model in a bounded smooth domain $\Omega\subset \mathbb R^3$. Assuming a smallness condition on the external force $f$, which does not necessitate smallness of $\| f\|_{[L^2(\Omega)]^3}$-norm, we show that for any smooth divergence free initial data $u_0$ there exists ${\mathcal T}={\mathcal T}(\|u_0\|_{[L^2(\Omega)]^3})$ satisfying $$ {\mathcal T} \to 0 \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to 0 \quad \text{and} \quad {\mathcal T} \to \infty \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to \infty, $$% and such that either a corresponding regular solution ceases to exist until $\mathcal T$ or, otherwise, it is globally defined and approaches a maximal compact invariant set $\mathbb A$. The latter set $\mathbb A$ is a global attractor for the semigroup restricted to initial velocities $u_0$ in a certain ball of fractional power space $X^{1/4}$ associated with the Stokes operator, which in turn does not necessitate smallness of the gradient norm $\|\nabla u_0\|_{[L^2(\Omega)]^3}$. Moreover, $\mathbb A$ attracts orbits of bounded sets in $X$ through Leray-Hopf type solutions obtained as limits of viscous parabolic approximations.

Uniqueness of the Leray-Hopf solution for a dyadic model

The dyadic problem $\dot u_n + \lambda^{2n} u_n - \lambda^{\beta n} u_{n-1}^2 + \lambda^{\beta(n+1)} u_n u_{n+1} = 0$ with "smooth" initial data is considered. The uniqueness of the Leray-Hopf solution is proved.

A Priori Bound on the Velocity in Axially Symmetric Navier–Stokes Equations

Let v be the velocity of Leray–Hopf solutions to the axially symmetric three-dimensional Navier–Stokes equations. Under suitable conditions for initial values, we prove the following a priori bound $$|v(x, t)| \le \frac{C |\ln r|^{1/2}}{r^2}, \qquad 0 < r \le 1/2,$$ where r is the distance from x to the z axis, and C is a constant depending only on the initial value. This provides a pointwise upper bound (worst case scenario) for possible singularities, while the recent papers (Chiun-Chuan et al., Commun PDE 34(1–3):203–232, 2009; Koch et al., Acta Math 203(1):83–105, 2009) gave a lower bound. The gap is polynomial order 1 modulo a half log term.

Remarks on the weak formulation of the Navier–Stokes equations on the 2D hyperbolic space

The Leray–Hopf solutions to the Navier–Stokes equation are known to be unique on R2. In our previous work, we showed the breakdown of uniqueness in a hyperbolic setting. In this article, we show how to formulate the problem in order so the uniqueness can be restored.