Leray Solutions Research Articles

Gluing Non-unique Navier–Stokes Solutions

We construct non-unique Leray solutions of the forced Navier-Stokes equations in bounded domains via gluing methods. This demonstrates a certain locality and robustness of the non-uniqueness discovered by the authors in [1].

Reflection of internal gravity waves in the form of quasi-axisymmetric beams

Preservation of the angle of reflection when an internal gravity wave hits a sloping boundary generates a focusing mechanism if the angle between the direction of propagation of the incident wave and the horizontal is close to the slope inclination (near-critical reflection). This paper provides an explicit description of the leading approximation of the unique Leray solution to the near-critical reflection of internal waves from a slope in the form of a beam wave. More precisely, our beam wave approach allows to construct a fully consistent and Lyapunov stable approximate solution, L2-close to the Leray solution, in the form of a beam wave, within a certain (nonlinear) time-scale. To the best of our knowledge, this is the first result where a mathematical study of internal waves in terms of spatially localized beam waves is performed.A beam wave is a linear superposition of rapidly oscillating plane waves, where the high frequency of oscillation is proportional to the inverse of a power of the small parameter measuring the weak amplitude of waves.Being localized in the physical space thanks to rapid oscillations (and high variations of the modulus of the wavenumber), beams are physically more relevant than plane waves/packets of waves, whose wavenumber is nearly fixed (microlocalized). At the mathematical level, this marks a strong difference between the previous plane waves/packets of waves analysis and our approach.The main novelty of this work is to exploit the spatial localization of beam waves to exhibit a spatially localized, physically relevant solution and to improve the previous mathematical results from a twofold perspective: 1) our beam wave approximate solution is the sum of a finite number of terms, each of them is a consistent solution to the system and there is no artificial/non-physical corrector; 2) thanks to the absence of artificial correctors (used in the previous results) and to the special structure of the nonlinear term, we can push the expansion of our solution to next orders, so improving the accuracy and enlarging the consistency time-scale.Finally, our results provide a set of initial conditions localized on rays, for which the Leray solution maintains approximately in L2 the same localization.

On the topological size of the class of Leray solutions with algebraic decay

AbstractIn 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303–313) provided an important result regarding the energy decay of Leray solutions to the incompressible Navier–Stokes in : if the associated Stokes flows had their norms bounded by for some , then the same would be true of . The converse also holds, as shown by Skalák (J. Math. Fluid Mech. 16 (2014) 431–446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier–Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two‐sided bounds of the form .

Non-uniqueness of Leray Solutions to the Hypodissipative Navier–Stokes Equations in Two Dimensions

We exhibit non-unique Leray solutions of the forced Navier-Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in \cite{albritton2021non}, the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as $t \to 0^+$. In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik \cite{Vishik1,Vishik2} to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in \cite{albritton2021non}.

LONG-TIME DECAY OF LERAY SOLUTION OF 3D-NSE WITH EXPONENTIAL DAMPING

We study the uniqueness, the continuity in [Formula: see text] and the large time decay for the Leray solutions of the 3D incompressible Navier–Stokes equations with nonlinear exponential damping term [Formula: see text], ([Formula: see text]).

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

In a seminal work, Leray (1934) demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a ``background" solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik (2018). The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Šverák (2015). Our solutions live precisely on the borderline of the known well-posedness theory.

A Note on Stokes Approximations to Leray Solutions of the Incompressible Navier–Stokes Equations in ℝn

In the early 1980s it was well established that Leray solutions of the unforced Navier–Stokes equations in Rn decay in energy norm for large t. With the works of T. Miyakawa, M. Schonbek and others it is now known that the energy decay rate cannot in general be any faster than t−(n+2)/4 and is typically much slower. In contrast, we show in this note that, given an arbitrary Leray solution u(·,t), the difference of any two Stokes approximations to the Navier–Stokes flow u(·,t) will always decay at least as fast as t−(n+2)/4, no matter how slow the decay of ∥u(·,t)∥L2(Rn) might be.

Stochastic Lagrangian Path for Leray’s Solutions of 3D Navier–Stokes Equations

In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution ${\mathbf u}$ of 3D-NSE and each $(s,x)\in\mathbb{R}_+\times\mathbb{R}^3$, we show the existence of weak solutions to the following SDE, which has a density $\rho_{s,x}(t,y)$ belonging to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u} (s,X_{s,t})\mathrm{d} t+\sqrt{2\nu}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$ where $W$ is a three dimensional standard Brownian motion, $\nu>0$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all $(s,x)$, the solution $X^n_{s,\cdot}(x)$ of the above SDE associated with the mollifying velocity field ${\mathbf u}_n$ weakly converges to $X_{s,\cdot}(x)$ so that $X$ is a Markov process in almost sure sense.

Leray's plane steady state solutions are nontrivial

We study solutions to the obstacle problem for the stationary Navier–Stokes system in a two dimensional exterior domain (flow past a prescribed body). We prove that the classical Leray solution to this problem is always nontrivial. No additional condition (on symmetry or smallness, etc.) is assumed. This is a complete extension of a classical result of C.J. Amick (1988) [1] where nontriviality was proved under symmetry assumption.

Some Liouville theorems for stationary Navier-Stokes equations in Lebesgue and Morrey spaces

Uniqueness of Leray solutions of the 3D Navier-Stokes equations is a challenging open problem. In this article we will study this problem for the 3D stationary Navier-Stokes equations in the whole space R3. Under some additional hypotheses, stated in terms of Lebesgue and Morrey spaces, we will show that the trivial solution U→=0 is the unique solution. This type of results are known as Liouville theorems.

Exponential convergence of mixed hp-DGFEM for the incompressible Navier–Stokes equations in ℝ2

Abstract In a polygon $\varOmega \subset \mathbb{R}^2$ we consider mixed $hp$-discontinuous Galerkin approximations of the stationary, incompressible Navier–Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and $hp$ spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in $\varOmega$, we prove exponential rates of convergence of the mixed $hp$-discontinuous Galerkin finite element method, with respect to the number of degrees of freedom, for small data which is piecewise analytic.

Global existence of a uniformly local energy solution for the incompressible fractional Navier–Stokes equations

We introduce the concept of local Leray solutions starting from locally square-integrable initial data to the fractional Navier–Stokes equations with $s\in [3/4,1)$. Furthermore, we prove their local-in-time existence when $s\in (3/4, 1)$. In particular,

Discretely self-similar solutions to the Navier–Stokes equations with data in Lloc2 satisfying the local energy inequality

Chae and Wolf recently constructed discretely self-similar solutions to the Navier-Stokes equations for any discretely self similar data in $L^2_{\mathrm{loc}}$. Their solutions are in the class of local Leray solutions with projected pressure, and satisfy the "local energy inequality with projected pressure". In this note, for the same class of initial data, we construct discretely self-similar suitable weak solutions to the Navier-Stokes equations that satisfy the classical local energy inequality of Scheffer and Caffarelli-Kohn-Nirenberg. We also obtain an explicit formula for the pressure in terms of the velocity. Our argument involves a new purely local energy estimate for discretely self-similar solutions with data in $L^2_{\mathrm{loc}}$ and an approximation of divergence free, discretely self-similar vector fields in $L^2_{\mathrm{loc}}$ by divergence free, discretely self-similar elements of $L^3_w$.

On two new inequalities for Leray solutions of the Navier-Stokes equations in [formula omitted

We derive two asymptotic inequalities, not previously observed, that are valid for spatial derivatives (of arbitrary order) of Leray solutions to the incompressible Navier-Stokes equations in Rn, n≤4. They are important to the decay investigation of solution derivatives.

An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes

We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier–Stokes on the torus , assuming that the solutions have norms for Besov space that are bounded in the L3-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form vanishing as if A consequence is that Onsager-type ‘quasi-singularities’ are required in the Leray solutions, even if the total energy dissipation vanishes in the limit , as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions u which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For the anomalous dissipation vanishes and the weak Euler solutions may be spatially ‘rough’ but conserve energy.

A Note on the Regularity Time of Leray Solutions to the Navier–Stokes Equations

In this note we slightly improve a well known result about the regularity time of Leray solutions $${{\varvec{u}}}(\cdot ,t)$$ to the Navier–Stokes equations in $$ \mathbb {R}^{n} \!\,\!$$ ( $$n \le 4$$ ). A related result on the eventual monotonicity of $$ \,\! \Vert \;\!D^{m}{{\varvec{u}}}(\cdot ,t) \,\!\Vert _{L^{2}(\mathbb {R}^{n})} \!\;\!$$ for arbitrary $$ m \,\!$$ is also discussed.

The Boltzmann Equation with Incoming Boundary Condition: Global Solutions and Navier--Stokes Limit

We consider the Boltzmann equations with cutoff collision kernels in bounded domains. For the initial data with finite physical bounds, we prove the existence of global-in-time renormalized solutions in the sense of DiPerna--Lions endowed with incoming boundary condition. Moreover, we justify the limit as the Knudsen number $\epsilon\rightarrow 0$ to Leray solutions of the incompressible Navier--Stokes--Fourier equations with homogeneous Dirichlet conditions from renormalized solutions of the scaled Boltzmann equations when the incoming data are close to the global Maxwellian in the sense of the boundary relative entropy of order $O(\epsilon^3)$.

Ill-Posedness of Leray Solutions for the Hypodissipative Navier–Stokes Equations

We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-\Delta)^\alpha$ with exponent $\alpha < \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and L\'aszl\'o Sz\'ekelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.

Will mathematicians unleash the power of the Navier-Stokes equations?

Since 1934 when Leray proved the global existence of finite energy weak solutions to 3D incompressible Navier-Stokes equations, the regularity and uniqueness of Leray weak solutions has been core in the mathematical theory of incompressible fluid mechanics. This problem is also closely related to the Millionaires problem proposed by Clay Institute, namely, the global regularity or singularity to finite energy solutions of 3D incompressible Navier-Stokes system. The goal of this article is first to introduce the problem and then to survey some progresses on this problem.In Section 1 , following Fefferman, we first present the Millionaires problem concerning 3D incompressible Navier-Stokes equations.We also present the result of Leray on the global existence of finite energy weak solutions to the system. We conclude the introduction byremarking that the reason why we cannot solve the global regularity of 3D Navier-Stokes is that the system is energy super-critical. Whereas the energy lawis so far the only conservation law which we can find for 3D Navier-Stokes system.The goal of Section 2 is to present the main approaches toward the regularity of 3D Navier-Stokes system, namely, partial regularityof suitable weak solutions to 3D Navier-Stokes system, conditional regularity of Leray solutions, and well-posedness of 3D Navier-Stokes system with initial datain the critical spaces.Finally, in the last section, we comment about the possible singularity of 3D Navier-Stokes system.

A Kato-type criterion for the zero-viscosity limit of the incompressible Navier-Stokes flows with vortex sheets data

There are a few examples of solutions to the incompressible Euler equations which are piecewise smooth with a discontinuity of the tangential velocity across a hypersurface evolving in time: the so-called vortex sheets. An important open problem is to determine whether or not these solutions can be obtained as zero viscosity limits of the incompressible Navier-Stokes solutions in the energy space. In this paper we establish a couple of sufficient conditions similar to the one obtained by Kato in [T.~Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. Seminar on nonlinear partial differential equations, 85-98, Math. Sci. Res. Inst. Publ., 2, 1984] for the convergence of Leray solutions to the Navier-Stokes equations in a bounded domain with no-slip condition towards smooth solutions to the Euler equation.