Conditions Implying Energy Equality for Weak Solutions of the Navier--Stokes Equations

When a Leray--Hopf weak solution to the NSE has a singularity set $S$ of dimension $d$ less than $3$---for example, a suitable weak solution---we find a family of new $L^q L^p$ conditions that guarantee validity of the energy equality. Our conditions surpass the classical Lions--Lady\v{z}enskaja $L^4 L^4$ result in the case $d<1$. Additionally, we establish energy equality in certain cases of Type-I blowup. The results are also extended to the NSE with fractional power of the Laplacian below $1$.Comment: Accepted version. 20 pages, 11 figure

Energy equality in compressible fluids with physical boundaries

We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, provided that the density is bounded and satisfies $\sqrt{\rho} \in L^\infty_t H^1_x$. The main idea is to construct a global mollification combined with an independent boundary cut-off, and then take a double limit to prove the convergence of the resolved energy

Three-dimensional shear driven turbulence with noise at the boundary

We consider the incompressible 3D Navier-Stokes equations subject to a shear induced by noisy movement of part of the boundary. The effect of the noise is quantified by upper bounds on the first two moments of the dissipation rate. The expected value estimate is consistent with the Kolmogorov dissipation law, recovering an upper bound as in [15] for the deterministic case. The movement of the boundary is given by an Ornstein-Uhlenbeck process; a potential for over-dissipation is noted if the Ornstein-Uhlenbeck process were replaced by the Wiener process.Comment: 22 pages, 1 figur

On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

We consider three-dimensional stochastically forced Navier-Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper and lower bounds of the mean value of the time-averaged energy dissipation rate, $\mathbb{E} [\langle\varepsilon \rangle]$, are derived directly from the equations. First, we show that for a weak (martingale) solution to the stochastically forced Navier-Stokes equations, $$\mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L},$$ where $G^2$ is the total energy rate supplied by the random force, $U$ is the root-mean-square velocity, $L$ is the longest length scale in the applied forcing function, and $Re$ is the Reynolds number. Under an additional assumption of energy equality, we also derive a lower bound if the energy rate given by the random force dominates the deterministic behavior of the flow in the sense that $G^2 > 2 F U$, where $F$ is the amplitude of the deterministic force. We obtain, $$\frac{1}{3} G^2 - \frac{1}{3} (2+ \frac{1}{Re})\frac{U^3}{L} \leq \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L}\,.$$ In particular, under such assumptions, we obtain the zeroth law of turbulence in the absence of the deterministic force as, $$\mathbb{E} [\langle\varepsilon \rangle] = \frac{1}{2} G^2.$$ Besides, we also obtain variance estimates of the dissipation rate for the model

The energy conservation and regularity for the Navier-Stokes equations

In this paper, we consider the energy conservation and regularity of the weak solution $u$ to the Navier-Stokes equations in the endpoint case. We first construct a divergence-free field $u(t,x)$ which satisfies $\lim_{t\to T}\sqrt{T-t}||u(t)||_{BMO}<\infty$ and $\lim_{t\to T}\sqrt{T-t}||u(t)||_{L^\infty}=\infty$ to demonstrate that the Type II singularity is admissible in the endpoint case $u\in L^{2,\infty}(BMO)$. Secondly, we prove that if a suitable weak solution $u(t,x)$ satisfying $||u||_{L^{2,\infty}([0,T];BMO(\Omega))}<\infty$ for arbitrary $\Omega\subseteq\mathbb{R}^3$ then the local energy equality is valid on $[0,T]\times\Omega$. As a corollary, we also prove $||u||_{L^{2,\infty}([0,T];BMO(\mathbb{R}^3))}<\infty$ implies the global energy equality on $[0,T]$. Thirdly, we show that as the solution $u$ approaches a finite blowup time $T$, the norm $||u(t)||_{BMO}$ must blow up at a rate faster than $\frac{c}{\sqrt{T-t}}$ with some absolute constant $c>0$. Furthermore, we prove that if $||u_3||_{L^{2,\infty}([0,T];BMO(\mathbb{R}^3))}=M<\infty$ then there exists a small constant $c_M$ depended on $M$ such that if $||u_h||_{L^{2,\infty}([0,T];BMO(\mathbb{R}^3))}\leq c_M$ then $u$ is regular on $(0,T]\times\mathbb{R}^3$

The Energy Measure for the Euler and Navier-Stokes Equations

The potential failure of energy equality for a solution $u$ of the Euler or Navier-Stokes equations can be quantified using a so-called `energy measure': the weak-$*$ limit of the measures |u(t)|^2\,\mbox{d}x as $t$ approaches the first possible blowup time. We show that membership of $u$ in certain (weak or strong) $L^q L^p$ classes gives a uniform lower bound on the lower local dimension of $\mathcal{E}$; more precisely, it implies uniform boundedness of a certain upper $s$-density of $\mathcal{E}$. We also define and give lower bounds on the `concentration dimension' associated to $\mathcal{E}$, which is the Hausdorff dimension of the smallest set on which energy can concentrate. Both the lower local dimension and the concentration dimension of $\mathcal{E}$ measure the departure from energy equality. As an application of our estimates, we prove that any solution to the $3$-dimensional Navier-Stokes Equations which is Type-I in time must satisfy the energy equality at the first blowup time.Comment: 26 pages, 3 figures. Accepted versio

On the energy equality for the 3D incompressible viscoelastic flows

In this paper, we study the problem of energy conservation for the solutions to the 3D viscoelastic flows. First, we consider Leray-Hopf weak solutions in the bounded convex domain $\Omega$. We prove that under the Shinbrot type conditions $u \in L^{q}_{loc}\left(0, T ; L^{p}(\Omega)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } {\bf F} \in L^{r}_{loc}\left(0, T ; L^{s}(\Omega)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4$, the boundary conditions $u|_{\partial\Omega}=0,{\bf F}\cdot n|_{\partial\Omega}=0$ can inhibit the boundary effect and guarantee the validity of energy equality. Next, we apply this idea to deal with the case $\Omega= \mathbb{R}^3$, and showed that the energy is conserved for $u\in L_{loc}^{q}\left(0,T;L_{loc}^{p}\left(\mathbb{R}^{3}\right)\right)$ with $\frac{2}{q}+\frac{2}{p}\leq1, p\geq 4$ and ${\bf F}\in L_{loc}^{r}\left(0,T;L_{loc}^{s}\left(\mathbb{R}^{3}\right)\right)\cap L^{\frac{20}{7}}\left(0,T;L^{\frac{20}{7}}\left(\mathbb{R}^{3}\right)\right)$ with $\frac{2}{r}+\frac{2}{s}\leq1, s\geq 4$. Our theorem shows that the behavior of solutions in the finite regions and the behavior at infinite play different roles in the energy conservation. Finally, we consider the problem of energy conservation for distributional solutions and show energy equality for the distributional solutions belonging to the so-called Lions class $L^4L^4$.Comment: arXiv admin note: substantial text overlap with arXiv:2108.1047

Anomalous dissipation, anomalous work, and energy balance for smooth solutions of the Navier-Stokes equations

In this paper, we study the energy balance for a class of solutions of the Navier-Stokes equations with external forces in dimensions three and above. The solution and force are smooth on $(0,T)$ and the total dissipation and work of the force are both finite. We show that a possible failure of the energy balance stems from two effects. The first is the \emph{anomalous dissipation} of the solution, which has been studied in many contexts. The second is what we call the \emph{anomalous work} done by the force, a phenomenon that has not been analyzed before. There are numerous examples of solutions exhibiting \emph{anomalous work}, for which even a continuous energy profile does not rule out the anomalous dissipation, but only implies the balance of the strengths of these two effects, which we confirm in explicit constructions. More importantly, we show that there exist solutions exhibiting \emph{anomalous dissipation} with zero \emph{anomalous work}. Hence the violation of the energy balance results from the nonlinearity of the solution instead of artifacts of the force. Such examples exist in the class $u \in L_t^{3 } B^{\frac{1}{3} -}_{3,\infty}$ and $f \in L_t^{2-} H^{-1}$, which implies the sharpness of many existing conditions on the energy balance.Comment: v2: revised version to appear in SIM

On the energy equality for the 3D Navier-Stokes equations

In this paper we study the problem of energy conservation for the solutions of the initial boundary value problem associated to the 3D Navier-Stokes equations, with Dirichlet boundary conditions. First, we consider Leray-Hopf weak solutions and we prove some new criteria, involving the gradient of the velocity. Next, we compare them with the existing literature in scaling invariant spaces and with the Onsager conjecture. Then, we consider the problem of energy conservation for very-weak solutions, proving energy equality for distributional solutions belonging to the so-called Shinbrot class. A possible explanation of the role of this classical class of solutions, which is not scaling invariant, is also given

Convex integration and phenomenologies in turbulence

In this review article we discuss a number of recent results concerning wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Sz\'ekelyhidi Jr., who extended Nash's fundamental ideas on $C^1$ flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name convex integration, have fundamental analogies the phenomenological theories of hydrodynamic turbulence. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. First, we give an elementary construction of nonconservative $C^{0+}_{x,t}$ weak solutions of the Euler equations, first proven by De Lellis-Sz\'ekelyhidi Jr.. Second, we present Isett's recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work of De Lellis-Sz\'ekelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class $C^{\frac 13-}_{x,t}$ are constructed, attaining any energy profile. Third, we give a concise proof of the authors' recent result, which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity class $C^0_t L^{2+}_x \cap C^0_t W^{1,1+}_x$. We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.Comment: 76 pages, 10 figures, minor correction