Relaxation of the incompressible porous media equation

László Székelyhidi Jr

Displaying similar documents to “Relaxation of the incompressible porous media equation”

$L^{q}$ -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...

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...

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 $1<q<2$ .

A uniqueness result for the continuity equation in two dimensions

Giovanni Alberti, Stefano Bianchini, Gianluca Crippa (2014)

Journal of the European Mathematical Society

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We characterize the autonomous, divergence-free vector fields $b$ on the plane such that the Cauchy problem for the continuity equation $\partial_{t} u + \frac{.}{\dot{}} (b u) = 0$ admits a unique bounded solution (in the weak sense) for every bounded initial datum; the characterization is given in terms of a property of Sard type for the potential $f$ associated to $b$ . As a corollary we obtain uniqueness under the assumption that the curl of $b$ is a measure. This result can be extended to certain non-autonomous vector fields $b$ with...

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.

A second order unconditionally positive space-time residual distribution method for solving compressible flows on moving meshes

Dobeš, Jiří, Deconinck, Herman

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A space-time formulation for unsteady inviscid compressible flow computations in 2D moving geometries is presented. The governing equations in Arbitrary Lagrangian-Eulerian formulation (ALE) are discretized on two layers of space-time finite elements connecting levels $n$ , $n + 1 / 2$ and $n + 1$ . The solution is approximated with linear variation in space (P1 triangle) combined with linear variation in time. The space-time residual from the lower layer of elements is distributed to the nodes at level...

On the condition of Λ-convexity in some problems of weak continuity and weak lower semicontinuity

Agnieszka Kałamajska (2001)

Colloquium Mathematicae

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We study the functional $I_{f} (u) = \int_{Ω} f (u (x)) d x$ , where u=(u₁, ..., uₘ) and each $u_{j}$ is constant along some subspace $W_{j}$ of ℝⁿ. We show that if intersections of the $W_{j}$ ’s satisfy a certain condition then $I_{f}$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_{j}}_{j = 1, . . ., m}$ to have the equivalence: $I_{f}$ is weakly continuous if and only if f is Λ-affine.

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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Generalization of the weak amenability on various Banach algebras

Madjid Eshaghi Gordji, Ali Jabbari, Abasalt Bodaghi (2019)

Mathematica Bohemica

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The generalized notion of weak amenability, namely $(ϕ, ψ)$ -weak amenability, where $ϕ, ψ$ are continuous homomorphisms on a Banach algebra $𝒜$ , was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(ϕ, ψ)$ -weak amenability on the measure algebra $M (G)$ , the group algebra $L^{1} (G)$ and the Segal algebra $S^{1} (G)$ , where $G$ is a locally compact group, are studied. As a typical example, the $(ϕ, ψ)$ -weak amenability of a special semigroup algebra is shown as well.

Local-in-time existence for the non-resistive incompressible magneto-micropolar fluids

Peixin Zhang, Mingxuan Zhu (2022)

Applications of Mathematics

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We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_{0} \in H^{s - 1 + ε}$ , $w_{0} \in H^{s - 1}$ and $b_{0} \in H^{s}$ for $s > \frac{3}{2}$ and any $0 < ε < 1$ . The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$ .

The subspace of weak $P$ -points of $ℕ^{*}$

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $W$ be the subspace of $ℕ^{*}$ consisting of all weak $P$ -points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$ -pseudocompact space for all $p \in ℕ^{*}$ .