Remark on regularity criterion for weak solutions to the shear thinning fluids

Jae-Myoung Kim

Displaying similar documents to “Remark on regularity criterion for weak solutions to the shear thinning fluids”

A regularity criterion for the 2D MHD and viscoelastic fluid equations

Zhuan Ye (2015)

Annales Polonici Mathematici

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This paper is dedicated to a regularity criterion for the 2D MHD equations and viscoelastic equations. We prove that if the magnetic field B, respectively the local deformation gradient F, satisfies $\nabla B, \nabla F \in L^{q} (0, T; L^{p} (ℝ ²))$ for 1/p + 1/q = 1 and 2 < p ≤ ∞, then the corresponding local solution can be extended beyond time T.

Interior $C^{1,\alpha}$ -Regularity of Weak Solutions to the Equations of Stationary Motions of Certain Non-Newtonian Fluids in Two Dimensions

Jorg Wolf (2007)

Bollettino dell'Unione Matematica Italiana

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In the present work we prove the interior Hölder continuity of the gradient matrix of any weak solution of equations, which describes the motion of non-Newtonian fluid in two dimensions, restricting ourself to the shear thinning case $1<q<2$ .

Time regularity of generalized Navier-Stokes equation with $p (x, t)$ -power law

Cholmin Sin (2023)

Czechoslovak Mathematical Journal

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We show time regularity of weak solutions for unsteady motion equations of generalized Newtonian fluids described by $p (x, t)$ -power law for $p (x, t) \geq (3 n + 2) / (n + 2)$ , $n \geq 2,$ by using a higher integrability property and fractional difference method. Moreover, as its application we prove that every weak solution to the problem becomes a local in time strong solution and that it is unique.

Pointwise regularity associated with function spaces and multifractal analysis

Stéphane Jaffard (2006)

Banach Center Publications

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The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces $C_{E}^{α} (x ₀)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E = L^{\infty}$ corresponds to the usual Hölder regularity,...

On the global regularity of $N$ -dimensional generalized Boussinesq system

Kazuo Yamazaki (2015)

Applications of Mathematics

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We study the $N$ -dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.

$C^{1, α}$ regularity for elliptic equations with the general nonstandard growth conditions

Sungchol Kim, Dukman Ri (2024)

Mathematica Bohemica

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We study elliptic equations with the general nonstandard growth conditions involving Lebesgue measurable functions on $Ω$ . We prove the global $C^{1, α}$ regularity of bounded weak solutions of these equations with the Dirichlet boundary condition. Our results generalize the $C^{1, α}$ regularity results for the elliptic equations in divergence form not only in the variable exponent case but also in the constant exponent case.

On weak minima of certain integral functionals

Gioconda Moscariello (1998)

Annales Polonici Mathematici

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We prove a regularity result for weak minima of integral functionals of the form $\int_{Ω} F (x, D u) d x$ where F(x,ξ) is a Carathéodory function which grows as ${| ξ |}^{p}$ with some p > 1.

Cauchy problem for the non-newtonian viscous incompressible fluid

Milan Pokorný (1996)

Applications of Mathematics

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We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor $τ^{V} (𝕖) = τ (𝕖) - 2 μ_{1} Δ 𝕖$ , where the nonlinear function $τ (𝕖)$ satisfies $τ_{i j} (𝕖) e_{i j} \geq c {| 𝕖 |}^{p}$ or $τ_{i j} (𝕖) e_{i j} \geq {c (| 𝕖 |}^{2} + {| 𝕖 |}^{p})$ . First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p > 1$ for both models. Then, under vanishing higher viscosity $μ_{1}$ , the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p > \frac{3 n}{n + 2}$ , its uniqueness...

Relaxation of the incompressible porous media equation

László Székelyhidi Jr (2012)

Annales scientifiques de l'École Normale Supérieure

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It was shown recently by Córdoba, Faraco and Gancedo in [1] that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework developed for the incompressible Euler equations in [4], uses ideas from the theory of laminates, in particular $T 4$ configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding $T 4$ configurations. We then use this to construct weak solutions to the unstable interface...

Uniform regularity for an isentropic compressible MHD- $P 1$ approximate model arising in radiation hydrodynamics

Tong Tang, Jianzhu Sun (2021)

Czechoslovak Mathematical Journal

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It is well known that people can derive the radiation MHD model from an MHD- $P 1$ approximate model. As pointed out by F. Xie and C. Klingenberg (2018), the uniform regularity estimates play an important role in the convergence from an MHD- $P 1$ approximate model to the radiation MHD model. The aim of this paper is to prove the uniform regularity of strong solutions to an isentropic compressible MHD- $P 1$ approximate model arising in radiation hydrodynamics. Here we use the bilinear commutator and...

$L^{q}$ -approach to weak solutions of the Oseen flow around a rotating body

Stanislav Kračmar, Šárka Nečasová, Patrick Penel (2008)

Banach Center Publications

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We consider the time-periodic Oseen flow around a rotating body in ℝ³. We prove a priori estimates in $L^{q}$ -spaces of weak solutions for the whole space problem under the assumption that the right-hand side has the divergence form. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional term -(ω ∧ x)·∇u + ω ∧ u in the equation of momentum where ω denotes the angular velocity. We prove the existence of generalized weak solutions...

Time regularity and functions of the Volterra operator

Zoltán Léka (2014)

Studia Mathematica

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Our aim is to prove that for any fixed 1/2 < α < 1 there exists a Hilbert space contraction T such that σ(T) = 1 and $| | T^{n + 1} - T ⁿ | | ≍ (n \geq 1)$ . This answers Zemánek’s question on the time regularity property.

Partial regularity of minimizers of quasiconvex integrals with subquadratic growth: the general case

Menita Carozza, Giuseppe Mingione (2001)

Annales Polonici Mathematici

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We prove partial regularity for minimizers of the functional $\int_{Ω} f (x, u (x), D u (x)) d x$ where the integrand f(x,u,ξ) is quasiconvex with subquadratic growth: $| f (x, u, ξ) | \leq L (1 + | ξ |^{p})$ , p < 2. We also obtain the same results for ω-minimizers.

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

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We study the Cauchy problem for the 3D MHD system with damping terms ${ε | u |}^{α - 1} u$ and ${δ | b |}^{β - 1} b$ (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.