Serrin Type Research Articles

Regularity criteria for the three-dimensional axially symmetric non-resistive incompressible magnetohydrodynamic system

In this paper, we study the three-dimensional axially symmetric non-resistive incompressible magnetohydrodynamic system. By establishing a rough blow-up criteria on ω θ r , we show a Ladyzhenskaya–Prodi–Serrin type regularity criteria with respect to the swirl component u θ of the velocity.

A weak-$$L^{p}$$ Prodi–Serrin type regularity criterion for the micropolar fluid equations in terms of the pressure

A weak-$$L^{p}$$ Prodi–Serrin type regularity criterion for the micropolar fluid equations in terms of the pressure

Global strong solutions of three-dimensional compressible non-isentropic micropolar fluid equations with far field vacuum

We are concerned with the Cauchy problem of three-dimensional (3D) compressible non-isentropic micropolar fluid equations with far field vacuum. With the help of Serrin type blow-up criterion, we show the global existence and uniqueness of strong solutions provided that the initial mass is suitably small. Our work generalizes the result of Huang et al. (2021) [8] to the case that vacuum is allowed at infinity.

On 3D generalized MHD equations with different dissipation exponents

We study the 3D generalized magnetohydrodynamics (gMHD) equations with dissipation terms −(−Δ)αu and −(−Δ)βb. It is proved that a weak solution (u,b) to gMHD equations is smooth on R3×(0,T] if u, ∇u or (−Δ)m/2u belongs to Lq(0,T;Lp(R3)) with p,q and m=min{α,β} satisfying the generalized Ladyzhenskaya–Prodi–Serrin type conditions.

Liouville theorem and qualitative properties of solutions for an integral system

In this paper, we are concerned with an integral system where . Base on the integrability of positive solutions, we obtain some Liouville theorems and the decay rates of positive solutions at infinity. In addition, we use the properties of the contraction map and the shrinking map to prove that is Lipschitz continuous. In particular, the Serrin type condition is established, which plays an important role to classify the positive solutions.

Nonlinear parabolic stochastic evolution equations in critical spaces part II

This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical L^2-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.

Regularity criterion for the 3D Boussinesq equations with partial viscous terms in the limiting cases

By means of Fourier analysis and the structure of the equation, for the 3D Boussinesq system with partial viscosity, we show the blow-up criterion of the smooth solution via a Beale–Kato–Majda type of supj∈Z∫0T‖Δ̇j∇×u‖∞dt and a Serrin type of ‖u‖C([0,T),B∞,∞−1) or especially its oscillation in the B∞,∞−1 norm with small amplitude.

Serrin–type regularity criteria for the 3D MHD equations via one velocity component and one magnetic component

In this paper, we consider the 3D MHD equations in which the viscosity coefficient \(\mu \) and the resistivity coefficient \(\nu \) are equal. We show that the Serrin–type conditions imposed on one component of the velocity \(u_{3}\) and one component of magnetic fields \(b_{3}\) with $$\begin{aligned} u_{3} \in L^{p_{0},1}(-1,0;L^{q_{0}}(B(2))),\ b_{3} \in L^{p_{1},1}(-1,0;L^{q_{1}}(B(2))), \end{aligned}$$\(\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1\) and \(3<q_{0},q_{1}<+\infty \), imply that the suitable weak solution is regular at (0, 0).

Local analysis of a two phase free boundary problem concerning mean curvature

We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative $\partial_n u$ is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some non-criticality condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreover, in the critical case, we employ the use of the Crandall-Rabinowitz theorem to show the existence of a branch of symmetry breaking solutions bifurcating from trivial ones. Finally, some remarks on the one phase case and a similar overdetermined problem of Serrin type are given.

Some Serrin type blow‐up criteria for the three‐dimensional viscous compressible flows with large external potential force

We provide a Serrin type blow‐up criterion for the 3‐D viscous compressible flows with large external potential force. For the Cauchy problem of the 3‐D compressible Navier–Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup‐norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.

On Conditional Regularity for the MHD Equations via Partial Components

In this paper we establish some new regularity criteria for the three dimensional incompressible MHD equations via partial components of velocity and magnetic fields. Moreover, some Prodi–Serrin type regularity conditions are obtained.

Optimality of Serrin type extension criteria to the Navier-Stokes equations

AbstractWe prove that a strong solutionuto the Navier-Stokes equations on (0,T) can be extended if eitheru∈Lθ(0,T;U˙∞,1/θ,∞−α$\begin{array}{} \displaystyle \dot{U}^{-\alpha}_{\infty,1/\theta,\infty} \end{array}$) for 2/θ+α= 1, 0 &lt;α&lt; 1 oru∈L2(0,T;V˙∞,∞,20$\begin{array}{} \displaystyle \dot{V}^{0}_{\infty,\infty,2} \end{array}$), whereU˙p,β,σs$\begin{array}{} \displaystyle \dot{U}^{s}_{p,\beta,\sigma} \end{array}$andV˙p,q,θs$\begin{array}{} \displaystyle \dot{V}^{s}_{p,q,\theta} \end{array}$are Banach spaces that may be larger than the homogeneous Besov spaceB˙p,qs$\begin{array}{} \displaystyle \dot{B}^{s}_{p,q} \end{array}$. Our method is based on a bilinear estimate and a logarithmic interpolation inequality.

Optimality of Serrin type extension criteria to the Navier-Stokes equations

We prove that a strong solution u to the Navier-Stokes equations on (0, T) can be extended if either u ∈ L θ (0, T; U˙ −α ∞,1/θ,∞) for 2/θ + α = 1, 0 < α < 1 or u ∈ L 2 (0, T; V˙ 0 ∞,∞,2 ) , where U˙ s p,β,σ and V˙ s p,q,θ are Banach spaces that may be larger than the homogeneous Besov space B˙ s p,q. Our method is based on a bilinear estimate and a logarithmic interpolation inequality.

Serrin type regularity criterion for the shear thinning fluids via the velocity field

By estimating the nonlinear term in an innovative way, we could obtain the optimal regularity criterion for the shear thinning fluids via the velocity field.

The global regularity for 3D inhomogeneous incompressible fluids with vacuum

In this paper, we investigate the global regularity for 3D inhomogeneous incompressible Navier–Stokes equations with vacuum. More precisely, we establish several Prodi–Serrin type regularity criteria in the Besov space in terms of only velocity or gradient of velocity which allow the initial density to contain vacuum. By using the embedding theory of Besov space, it is shown that our results improve the Prodi–Serrin type regularity condition which is scaling invariant in the time–space Lebesgue space.

The second-order parabolic PDEs with singular coefficients and applications

The goal of this paper is to establish the Lipschitz and estimates for a second-order parabolic PDE on with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev differentiability of strong solutions to a class of SDEs with singular drift coefficients.

Blowup Mechanism for a Fluid-Particle Interaction System in $\mathbb{R}^{3}$

We study the Cauchy problem and the mixed initial boundary value problem of a fluid-particle interaction system in $\mathbb{R}^{3}$ . A Serrin type criterion for the strong solution of the Cauchy problem is established in terms of $\|\rho \|_{L^{\infty }_{t}L^{\infty }_{x}}$ and $\|u\|_{L^{s}_{t}L^{r}_{x}}$ , where $2/s+3/r\le 1$ and $3< r\le \infty $ . In view of some useful integral inequalities, we prove the life span estimates of the regular solution.

On Brunn–Minkowski Type Inequalities and Overdetermined Problem for p-Capacity

In this paper, we mainly study some properties of p-capacity. Firstly, the p-capacitary difference Brunn–Minkowski and Minkowski inequalities are established. Secondly, we propose the p-capacitary overdetermined problem and prove the Serrin type symmetry result for the problem. Finally, we also make some considerations for the polar set of p-difference body for capacity.

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.

Overdetermined problems and constant mean curvature surfaces in cones

We consider a partially overdetermined problem in a sector-like domain \Omega in a cone \Sigma in \mathbb{R}^N , N\geq 2 , and prove a rigidity result of Serrin type by showing that the existence of a solution implies that \Omega is a spherical sector, under a convexity assumption on the cone. We also consider the related question of characterizing constant mean curvature compact surfaces \Gamma with boundary which satisfy a 'gluing' condition with respect to the cone \Sigma . We prove that if either the cone is convex or the surface is a radial graph then \Gamma must be a spherical cap. Finally we show that, under the condition that the relative boundary of the domain or the surface intersects orthogonally the cone, no other assumptions are needed.