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

Kazuo Yamazaki

Displaying similar documents to “On the global regularity of $N$ -dimensional generalized Boussinesq system”

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,...

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

Jae-Myoung Kim (2024)

Mathematica Bohemica

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J. Q. Yang (2019) established a regularity criterion for the 3D shear thinning fluids in the whole space $ℝ^{3}$ via two velocity components. The goal of this short note is to extend this result in viewpoint of Lorentz space.

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.

$C^{1, α}$ 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.

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.

Regularity of solutions of the fractional porous medium flow

Luis Caffarelli, Fernando Soria, Juan Luis Vázquez (2013)

Journal of the European Mathematical Society

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We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. The precise model is $u_{t} = \nabla \cdot (u \nabla {(- Δ)}^{- s} u), 0 < s < 1$ . The problem is posed in ${x \in ℝ^{n}, t \in ℝ}$ with nonnegative initial data $u (x, 0)$ that are integrable and decay at infinity. A previous paper has established the existence of mass-preserving, nonnegative weak solutions satisfying energy estimates and finite propagation. As main results we establish the boundedness and $C^{α}$ regularity of such weak solutions. Finally, we extend...

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.

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.

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.

Global well-posedness for the Klein-Gordon-Schrödinger system with higher order coupling

Agus Leonardi Soenjaya (2022)

Mathematica Bohemica

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Global well-posedness for the Klein-Gordon-Schrödinger system with generalized higher order coupling, which is a system of PDEs in two variables arising from quantum physics, is proven. It is shown that the system is globally well-posed in $(u, n) \in L^{2} \times L^{2}$ under some conditions on the nonlinearity (the coupling term), by using the $L^{2}$ conservation law for $u$ and controlling the growth of $n$ via the estimates in the local theory. In particular, this extends the well-posedness results for such a system in...

Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther (2002)

Colloquium Mathematicae

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We consider the maximal function $| | (S^{a} f) [x] {| |}_{L^{\infty} [- 1, 1]}$ where $(S^{a} f) {(t)}^{\land} (ξ) = e^{{i t | ξ |}^{a}} f ̂ (ξ)$ and 0 < a < 1. We prove the global estimate $| | S^{a} f {| |}_{L ² (ℝ, L^{\infty} [- 1, 1])} {\leq C | | f | |}_{H^{s} (ℝ)}$ , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality

Mark A. Peletier, Robert Planqué, Matthias Röger (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1, 2} (ℝ^{n})$ . This criterion consists of comparing the value of a functional $\int f (u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$ , and we prove that the rate of convergence to zero is connected to regularity: $u \in W^{1, 2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to...

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p, q)$ -growth with exponents $p \leq q < \infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1, μ}$ . Note that to our knowledge the only $C^{1, μ}$ -results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

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Let Y be a Banach space and let $S \subset L_{p}$ be a subspace of an $L_{p}$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S (Y) \subset L_{p} (Y)$ . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{- t B}$ is a positive contraction...

Maximal regularity of discrete and continuous time evolution equations

Sönke Blunck (2001)

Studia Mathematica

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We consider the maximal regularity problem for the discrete time evolution equation $u_{n + 1} - T u ₙ = f ₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${(T ⁿ)}_{n \in ℕ ₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent...

Partial Boundary Regularity of Solutions of Nonlinear Superelliptic Systems

Christoph Hamburger (2007)

Bollettino dell'Unione Matematica Italiana

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We prove global partial regularity of weaksolutions of the Dirichlet problem for the nonlinear superelliptic system $\operatorname{div}A(x,u,Du)+B(x,u,DU)=0$ , under natural polynomial growth of the coefficient functions $A$ and $B$ . We employ the indirect method of the bilinear form and do not use a Caccioppoli or a reverse Hölder inequality.

Attractor of a semi-discrete Benjamin-Bona-Mahony equation on ℝ¹

Chaosheng Zhu (2015)

Annales Polonici Mathematici

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This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank-Nicolson scheme to discretize the Benjamin-Bona-Mahony equation on ℝ¹. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in H¹(ℝ¹). Then we prove that this system possesses a global attractor $_{τ}$ in H¹(ℝ¹). In addition, we show that the global attractor $_{τ}$ is regular, i.e., $_{τ}$ ...

Maximal regularity for second order non-autonomous Cauchy problems

Charles J. K. Batty, Ralph Chill, Sachi Srivastava (2008)

Studia Mathematica

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We consider some non-autonomous second order Cauchy problems of the form ü + B(t)u̇ + A(t)u = f(t ∈ [0,T]), u(0) = u̇(0) = 0. We assume that the first order problem u̇ + B(t)u = f(t ∈ [0,T]), u(0) = 0, has $L^{p}$ -maximal regularity. Then we establish $L^{p}$ -maximal regularity of the second order problem in situations when the domains of B(t₁) and A(t₂) always coincide, or when A(t) = κB(t).

A new look at an old comparison theorem

Jaroslav Jaroš (2021)

Archivum Mathematicum

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We present an integral comparison theorem which guarantees the global existence of a solution of the generalized Riccati equation on the given interval $[a, b)$ when it is known that certain majorant Riccati equation has a global solution on $[a, b)$ .

Global Attractor for the Convective Cahn-Hilliard Equation in $H^{k}$

Xiaopeng Zhao, Ning Duan (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider the convective Cahn-Hilliard equation with periodic boundary conditions. Based on the iteration technique for regularity estimates and the classical theorem on existence of a global attractor, we prove that the convective Cahn-Hilliard equation has a global attractor in $H^{k}$ .

Displaying similar documents to “On the global regularity of N -dimensional generalized Boussinesq system”

Displaying similar documents to “On the global regularity of $N$ -dimensional generalized Boussinesq system”