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Yanqing Wang

Publications and source records attributed to Yanqing Wang.

At least 37 records · Page 2Linked to original sources
arXiv2022

Energy and helicity conservation for the generalized quasi-geostrophic equation

In this paper, we consider the 2-D generalized surface quasi-geostrophic equation with the velocity $v$ determined by $v=\mathcal{R}^{\perp}Λ^{γ-1}θ$. It is shown that the $L^p$ type energy norm of weak solutions is conserved provided $θ\in L^{p+1}(0,T; {B}^{\fracγ{3}}_{p+1, c(\mathbb{N})})$ for $0<γ<\frac32$ or $θ\in L^{p+1}(0,T; {B}^α_{p+1,\infty})~\text{for any}~γ-1<α<1 \text{ with} ~\frac{3}{2}\leq γ<2$. Moreover, we also prove that the helicity of weak solutions satisfying $\nablaθ\in L^{3}(0,T;\dot{B}_{3,c(\mathbb{N})}^{\fracγ{3}})$ for $0<γ<\frac32$ or $\nablaθ\in L^{3}(0,T; \dot{B}^α_{3,\infty})~\text{for any}~γ-1<α<1 \text{ with} ~\frac{3}{2}\leq γ<2$ is invariant. Therefore, the accurate relationships between the critical regularity for the energy (helicity) conservation of the weak solutions and the regularity of velocity in 2-D generalized quasi-geostrophic equation are presented.

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arXiv2022

Fractal dimension of potential singular points set in the Navier-Stokes equations under supercritical regularity

The main objective of this paper is to answer the questions posed by Robinson and Sadowski [21, p. 505, Comm. Math. Phys., 2010]{[RS3]} for the Navier-Stokes equations. Firstly, we prove that the upper box dimension of the potential singular points set $\mathcal{S}$ of suitable weak solution $u$ belonging in $ L^{q}(0,T;L^{p}(\mathbb{R}^{3}))$ for $1\leq\frac{2}{q}+\frac{ 3}{p}\leq\frac32$ with $2\leq q<\infty$ and $2<p<\infty$ is at most $\max\{p,q\}(\frac{2}{q}+\frac{ 3}{p}-1)$ in this system. Secondly, it is shown that $1-2 s$ dimension Hausdorff measure of potential singular points set of suitable weak solutions satisfying $ u\in L^{2}(0,T;\dot{H}^{s+1}(\mathbb{R}^{3}))$ for $0\leq s\leq\frac12$ is zero, whose proof relies on Caffarelli-Silvestre's extension. Inspired by Baker-Wang's recent work [1], this further allows us to discuss the Hausdorff dimension of potential singular points set of suitable weak solutions if the gradient of the velocity under some supercritical regularity.

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arXiv2022

Energy conservation of weak solutions for the incompressible Euler equations via vorticity

Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the $L^p$ control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion $ω=\text{curl}u\in L^{3}(0,T;L^{\frac{3n}{n+2}}(Ω))\,(n=2,3)$, which generalizes the corresponding result in [8] and can be viewed as in Onsager critical spatio-temporal spaces. Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and $\nabla\sqrtρ$ belongs to $L^{\infty}(0,T;L^{n}(\mathbb{T}^{n}))\,(n=2,3)$, which gives an affirmative answer to a problem proposed by Chen and Yu in [5].

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arXiv2022

Analytical validation of the helicity conservation for the compressible Euler equations

In [25], Moffatt introduced the concept of helicity in an inviscid fluid and examined the helicity preservation of smooth solution to barotropic compressible flow. In this paper, it is shown that the weak solutions of the above system in Onsager type spaces $\dot{B}^{1/3}_{p,c(\mathbb{N})}$ guarantee the conservation of the helicity. The parallel results of homogeneous incompressible Euler equations and the surface quasi-geostrophic equation are also obtained.

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arXiv2022

Partial regularity of suitable weak solutions of the model arising in amorphous molecular beam epitaxy

In this paper, we are concerned with the precise relationship between the Hausdorff dimension of possible singular point set $\mathcal{S}$ of suitable weak solutions and the parameter $α$ in the nonlinear term in the following parabolic equation $$h_t+h_{xxxx}+\partial_{xx}|h_x|^α=f.$$ It is shown that when $5/3\leqα<7/3$, the $\frac{3α-5}{α-1}$-dimensional parabolic Hausdorff measure of $\mathcal{S}$ is zero, which generalizes the recent corresponding work of Ozánski and Robinson in [31,SIAM J. Math. Anal. 51: 228--255, 2019] for $α=2$ and $f=0$. The same result is valid for a 3D modified Navier-Stokes system.

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arXiv2022

A general sufficient criterion for energy conservation in the Navier-Stokes system

In this paper, we derive an energy conservation criterion based on a combination of velocity and its gradient for the weak solutions of both the homogeneous incompressible Navier-Stokes equations and the general compressible Navier-Stokes equations. For the incompressible case, this class implies most known corresponding results on periodic domain via either the velocity or its gradient including the famous Lions' energy conservation criterion obtained in \cite{[Lions]}. For the compressible case, this helps us to extend the previously known criteria for the energy conservation of weak solutions from the incompressible fluid to compressible flow and improve the recent results due to Nguyen-Nguyen-Tang in \cite[Nonlinearity 32 (2019)]{[NNT]} and Liang in \cite[Proc. Roy. Soc. Edinburgh Sect. A (2020)]{[Liang]}.

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arXiv2021

Energy equality for the isentropic compressible Navier-Stokes equations without upper bound of the density

In this paper, we are concerned with the minimal regularity of both the density and the velocity for the weak solutions keeping energy equality in the isentropic compressible Navier-Stokes equations. The energy equality criteria without upper bound of the density are established. Almost all previous corresponding results requires $ρ\in L^{\infty}(0,T;L^{\infty}(\mathbb{T}^{d}))$.

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arXiv2021

The role of density in the energy conservation for the isentropic compressible Euler equations

In this paper, we study Onsager's conjecture on the energy conservation for the isentropic compressible Euler equations via establishing the energy conservation criterion involving the density $\varrho\in L^{k}(0,T;L^{l}(\mathbb{T}^{d}))$. The motivation is to analysis the role of the integrability of density of the weak solutions keeping energy in this system, since almost all known corresponding results require $\varrho\in L^{\infty}(0,T;L^{\infty}(\mathbb{T}^{d}))$. Our results imply that the lower integrability of the density $\varrho$ means that more integrability of the velocity $v$ are necessary in energy conservation and the inverse is also true. The proof relies on the Constantin-E-Titi type and Lions type commutators on mollifying kernel.

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arXiv2021

Gagliardo-Nirenberg inequalities in Lorentz type spaces and energy equality for the Navier-Stokes system

In this paper, we derive some new Gagliardo-Nirenberg type inequalities in Lorentz type spaces without restrictions on the second index of Lorentz norms, which generalize almost all known corresponding results. Our proof mainly relies on the Bernstein inequalities in Lorentz spaces, the embedding relation among various Lorentz type spaces, and Littlewood-Paley decomposition techniques. In addition, we establish several novel criteria in terms of the velocity or the gradient of the velocity in Lorentz spaces for energy conservation of the 3D Navier-Stokes equations. Particularly, we improve the classical Shinbrot's condition for energy balance to allow both the space-time directions of the velocity to be in Lorentz spaces.

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arXiv2021

Numerics for Stochastic Distributed Parameter Control Systems: a Finite Transposition Method

In this chapter, we present some recent progresses on the numerics for stochastic distributed parameter control systems, based on the \emph{finite transposition method} introduced in our previous works. We first explain how to reduce the numerics of some stochastic control problems in this respect to the numerics of backward stochastic evolution equations. Then we present a method to find finite transposition solutions to such equations. At last, we give an illuminating example.

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arXiv2020

Strong Rates of Convergence for Space-Time Discretization of the Backward Stochastic Heat Equation, and of a Linear-Quadratic Control Problem for the Stochastic Heat Equation

We introduce a time-implicit, finite-element based space-time discretization scheme for the backward stochastic heat equation, and for the forward-backward stochastic heat equation from stochastic optimal control, and prove strong rates of convergence. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise.

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arXiv2020

Strong Error Estimates for a Space-Time Discretization of the Linear-Quadratic Control Problem with the Stochastic Heat Equation with Linear Noise

We propose a time-implicit, finite-element based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+σ(t)]dW(t)$, and prove optimal convergence w.r.t. both, space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work [16], where the error analysis was restricted to additive noise.

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arXiv2020

Leray's backward self-similar solutions to the 3D Navier-Stokes equations in Morrey spaces

In this paper, it is shown that there does not exist a non-trivial Leray's backward self-similar solution to the 3D Navier-Stokes equations with profiles in Morrey spaces $\dot{\mathcal{M}}^{q,1}(\mathbb{R}^{3})$ provided $3/2 3/2$, and Guevara-Phuc [11, SIAM J. Math. Anal. 12 (2018)] in $\dot{\mathcal{M}}^{q,\frac{12-2q}{3}}(\mathbb{R}^{3})$ with $12/5\leq q<3$ and in $L^{q, \infty}(\mathbb{R}^3)$ with $12/5\leq q<6$.

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arXiv2019

On continuation criteria for the full compressible Navier-Stokes equations in Lorentz spaces

In this paper, we derive several new sufficient conditions of non-breakdown of strong solutions for for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant $\varepsilon$ such that the solution $(ρ,u,θ)$ to full compressible Navier-Stokes equations can be extended beyond $t=T$ provided that one of the following two conditions holds (1) $ρ\in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$, $u\in L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))$ and $$\| u\|_{L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))}\leq \varepsilon, ~~\text{with}~~ {2/p}+ {3/q}=1,\ \ q>3;$$ (2) $λ<3μ,$ $ρ\in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$, $θ\in L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))$ and $$\|θ\|_{L^{p,\infty}(0,T; L^{q,\infty}(\mathbb{R}^{3}))}\leq \varepsilon, ~~\text{with}~~ {2/p}+ {3/q}=2,\ \ q>3/2.$$ To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces to the full Navier-Stokes system. Third, without the condition on $ρ$ in (0.1) and (0.3), the results also hold for the 3D nonhomogeneous incompressible Navier-Stokes equations. The appearance of vacuum in these systems could be allowed.

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arXiv2019

Refined blow up criteria for the full compressible Navier-Stokes equations involving temperature

In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier-Stokes equations. Enlightening regular criteria via pressure $Π=\frac{\text {divdiv}}{-Δ}(u_{i}u_{j})$ of the 3D incompressible Navier-Stokes equations on bounded domain, we generalize Beirao da Veiga's result in [1] from the incompressible Navier-Stokes equations to the isentropic compressible Navier-Stokes system in the case away from vacuum.

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arXiv2019

$\varepsilon$-regularity criteria in Lorentz spaces to the 3D Navier-Stokes equations

In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Leray's blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,\infty}(-1,0;L^{q,l}(\mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<\infty$ and $q\leq l <\infty$ is finite.

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