Prodi Serrin Condition Research Articles

A note on weak existence for singular SDEs

Recently Krylov [N. V. Krylov, On time inhomogeneous stochastic Itô equations with drift in [Formula: see text], Ukraïn. Mat. Zh. 72 (2020) 1232–1253] established weak existence of solutions to SDEs for integrable drifts in mixed Lebesgue spaces, whose exponents satisfy the condition [Formula: see text], thus going below the celebrated Ladyzhenskaya–Prodi–Serrin condition. We present here a variant of such result, whose proof relies on an alternative technique, based on a partial Zvonkin transform; this allows for drifts with growth at infinity and/or in uniformly local Lebesgue spaces.

Uniqueness and regularity of weak solutions of a fluid-rigid body interaction system under the Prodi-Serrin condition

Uniqueness and regularity of weak solutions of a fluid-rigid body interaction system under the Prodi-Serrin condition

Anisotropic Prodi–Serrin regularity criteria for the 3D Navier–Stokes equations involving the gradient of one velocity component

In this paper, we prove some optimal regularity criteria for the 3D incompressible Navier–Stokes equations involving the gradient of one velocity component in the framework of anisotropic Lebesgue spaces satisfying Prodi–Serrin condition. These results cover the previous ones established by Wolf (2015) and O (2023).

Searching for Singularities in Navier–Stokes Flows Based on the Ladyzhenskaya–Prodi–Serrin Conditions

In this investigation we perform a systematic computational search for potential singularities in 3D Navier-Stokes flows based on the Ladyzhenskaya-Prodi-Serrin conditions. They assert that if the quantity $\int_0^T \| \mathbf{u}(t) \|_{L^q(\Omega)}^p \, dt$, where $2/p+3/q \le 1$, $q > 3$, is bounded, then the solution $\mathbf{u}(t)$ of the Navier-Stokes system is smooth on the interval $[0,T]$. In other words, if a singularity should occur at some time $t \in [0,T]$, then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions $\mathbf{u}_0$ are sought to maximize $\int_0^T \| \mathbf{u}(t) \|_{L^4(\Omega)}^8 \, dt$ for different $T$ subject to suitable constraints. These problems are solved numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$. However, the maximum enstrophy attained in these extreme flows scales in proportion to $\mathcal{E}_0^{3/2}$, the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of $\| \mathbf{u}(t) \|_{L^4(\Omega)}$ by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could be improved.

Prodi–Serrin condition for 3D Navier–Stokes equations via one directional derivative of velocity

In this paper, we consider the conditional regularity of weak solution to the 3D Navier–Stokes equations. More precisely, we prove that if one directional derivative of velocity, say ∂ 3 u , satisfies ∂ 3 u ∈ L p 0 , 1 ( 0 , T ; L q 0 ( R 3 ) ) with 2 p 0 + 3 q 0 = 2 and 3 2 < q 0 < + ∞ , then the weak solution is regular on ( 0 , T ] . The proof is based on the new local energy estimates introduced by Chae-Wolf (2019) [4] and Wang-Wu-Zhang (2020) [21] .

On a two components condition for regularity of the 3D Navier–Stokes equations under physical slip boundary conditions on non-flat boundaries

This work concerns the sufficient condition for the regularity of solutions to the evolution Navier–Stokes equations known in the literature as Prodi–Serrin condition. H.-O. Bae and H. J. Choe proved in 1997 that, in the whole space $$\mathbb {R}^3,$$ it is sufficient that two components of the velocity satisfy the above condition in order to guarantee the regularity of solutions. In 2017, H. Beirao da Veiga extended this result (Beirao da Veiga, J Math Anal Appl 453:212–220, 2017) to the half-space case $$\mathbb {R}^n_+$$ under slip boundary conditions by assuming that the velocity components parallel to the boundary enjoy the above condition. It remained open whether the flat boundary geometry is essential. Below, we prove that, under physical slip boundary conditions imposed in cylindrical boundaries, the result still holds.

Quantitative stability estimates for Fokker–Planck equations

We derive quantitative stability estimates for solutions of Fokker–Planck equations with irregular coefficients. We are mainly concerned with two different situations: in the degenerate case, the coefficients are assumed to be weakly differentiable, while in the non-degenerate case the drift coefficient satisfies only the Ladyzhenskaya–Prodi–Serrin condition. Our method is based on Trevisan's superposition principle, which represents the solution to the Fokker–Planck equation as the marginal distribution of the martingale solution of the associated stochastic differential equation.

Weak and Strong Solutions of the 3D Navier–Stokes Equations and Their Relation to a Chessboard of Convergent Inverse Length Scales

Using the scale invariance of the Navier-Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the $3D$ Navier-Stokes equations, an infinite `chessboard' of estimates for these inverse length scales is displayed in terms of labels $(n,\,m)$ corresponding to $n$ derivatives of the velocity field in $L^{2m}$. The $(1,\,1)$ position corresponds to the inverse Kolmogorov length $Re^{3/4}$. These estimates ultimately converge to a finite limit, $Re^3$, as $n,\,m\to \infty$, although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by $(n,\,m)$. In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by $(n,\,m)$, the only difference being a factor of 2 in the exponent. This appears to be a generalisation of the Prodi-Serrin conditions for $n\geq 1$.

Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows

We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier–Stokes equations with evolutionary equations for the deformation gradient and for the magnetization obtained from a special case of the micromagnetic energy. It turns out that the conditions on uniqueness coincide with those for the well-known Navier–Stokes equations in bounded domains: weak solutions are unique in two spatial dimensions, and weak solutions satisfying the Prodi–Serrin conditions are unique among all weak solutions in three dimensions. That is, we obtain the so-called weak-strong uniqueness result in three spatial dimensions.

A remark on the global regularity criterion for the 3D Navier–Stokes equations based on end-point Prodi–Serrin conditions

By taking full advantage of the regularity of the vertical velocity component, we could be able to show a regularity criterion in terms of the vertical velocity component, and the vertical derivative of the horizontal velocity components, which extends (Qian, 2016) significantly.

Some problems on the regularity of solutions for the 3D Navier-Stokes equations

In this paper, we review some recent work on the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations in recent years. It is well known that the three-dimensional incompressible Navier-Stokes system has the global Leray-Hopf weak solutions. When the weak solution satisfies the Prodi-Serrin condition, the solution is regular. We obtain some new results in the regularity conditions. Especially, for axisymmetric system, when the rotation speed is zero, it is well known that the system is globally well-posed. We find a new conserved quantity, and obtain some new advances in the regularity conditions of axisymmetric solutions with nonzero swirl, also get the global well-posedness results with the small initial rotating speed. Furthermore, we also get the similar result for the inhomogeneous system. In the end, we consider the global well-posedness of a class of generalized Navier-Stokes systems with the higher order horizontal viscosity $D_h^{2\alpha}$, $\alpha\geq\frac43$.

A Note on Prodi–Serrin Conditions for the Regularity of a Weak Solution to the Navier–Stokes Equations

The paper is concerned with the regularity of weak solutions to the Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin condition in order to obtain regularity for t > 0. The most interesting aspect of the result is that no compatibility condition is required to the initial data $v_0\in J^2(\OO) J2({\Omega})$.

On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier–Stokes equations. The half-space case

This note concerns the sufficient condition for regularity of solutions to the evolution Navier–Stokes equations known in the literature as Prodi–Serrin's condition. H.-O. Bae and H.J. Choe proved in a 1997 paper that, in the whole space R3, it is merely sufficient that two components of the velocity satisfy the above condition. Below, we extend the result to the half-space case R+n under slip boundary conditions. We show that it is sufficient that the velocity component parallel to the boundary enjoys the above condition. Flat boundary geometry is not essential, as shown in a forthcoming paper in cylindrical domains, prepared in collaboration with J. Bemelmans and J. Brand.

Non blow-up criterion for the 3-D Magneto-hydrodynamics equations in the limiting case

In this paper, we prove that suitable weak solution (u, b) of the 3-D MHD equations can be extended beyond T if u ∈ L ∞(0, T;L 3(ℝ3)) and the horizontal components b h of the magnetic field satisfies the well-known Ladyzhenskaya–Prodi–Serrin condition, which improves the corresponding regularity criterion by Mahalov–Nicolaenko–Shilkin.

Uniqueness Results for Weak Leray\u2013Hopf Solutions of the Navier\u2013Stokes System with Initial Values in Critical Spaces

The main subject of this paper concerns the establishment of certain classes of initial data, which grant short time uniqueness of the associated weak Leray–Hopf solutions of the three dimensional Navier–Stokes equations. In particular, our main theorem that this holds for any solenodial initial data, with finite L_2(mathbb {R}^3) norm, that also belongs to certain subsets of {textit{VMO}}^{-1}(mathbb {R}^3). As a corollary of this, we obtain the same conclusion for any solenodial u_{0} belonging to L_{2}(mathbb {R}^3)cap mathbb {dot{B}}^{-1+frac{3}{p}}_{p,infty }(mathbb {R}^3), for any 3<p<infty . Here, mathbb {dot{B}}^{-1+frac{3}{p}}_{p,infty }(mathbb {R}^3) denotes the closure of test functions in the critical Besov space {dot{B}}^{-1+frac{3}{p}}_{p,infty }(mathbb {R}^3). Our results rely on the establishment of certain continuity properties near the initial time, for weak Leray–Hopf solutions of the Navier–Stokes equations, with these classes of initial data. Such properties seem to be of independent interest. Consequently, we are also able to show if a weak Leray–Hopf solution u satisfies certain extensions of the Prodi-Serrin condition on mathbb {R}^3 times ]0,T[, then it is unique on mathbb {R}^3 times ]0,T[ amongst all other weak Leray–Hopf solutions with the same initial value. In particular, we show this is the case if uin L^{q,s}(0,T; L^{p,s}(mathbb {R}^3)) or if it’s L^{q,infty }(0,T; L^{p,infty }(mathbb {R}^3)) norm is sufficiently small, where 3<p< infty , 1le s<infty and 3/p+2/q=1.

Wellposedness for stochastic continuity equations with Ladyzhenskaya–Prodi–Serrin condition

We consider the stochastic divergence-free continuity equations with Ladyzhenskaya–Prodi–Serrin condition. Wellposedness is proved meanwhile uniqueness may fail for the deterministic PDE. The main issue of strong uniqueness, in the probabilistic sense, relies on stochastic characteristic method and the generalized Ito–Wentzell–Kunita formula. The stability property for the unique solution is proved with respect to the initial data. Moreover, a persistence result is established by a representation formula.

New version of the Ladyzhenskaya–Prodi–Serrin condition

New version of the Ladyzhenskaya–Prodi–Serrin condition

New Conditions for Local Regularity of a Suitable Weak Solution to the Navier--Stokes Equation

We formulate conditions which guarantee that a suitable weak solution (v,p) to the Navier—Stokes equation (in the sense of L. Caffarelli, R. Kohn and L. Nirenberg [2]) cannot have a singularity at the point (x0, t0). The usual Prodi—Serrin condition on velocity v is substantially replaced by an analogous condition imposed on the negative part p_ of pressure p.