Existence of weak solutions to doubly degenerate diffusion equations

Matas, Aleš; Merker, Jochen

doi:10.1007/s10492-012-0004-0

Cited by 8 publications

(10 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, and the proof that weak limits and expected limits are identical (e.g. (Bu) ex = Bu) -are in complete analogy with the proof for the prototypical case in [10]. Thus, under the assumptions (A1)-(A4) existence of weak solutions of the doubly nonlinear parabolic equation 1in the sense of Definition 1.3 can be proved for initial values u 0 ∈ Y .…”

Section: Consider the Right Hand Side As An Operatormentioning

confidence: 62%

“…In fact, similar to [14, 3.2] in Section 2 it is shown that the approximate equation obtained by restricting (1) to a finite-dimensional subspace can be solved under these assumptions with the help of the Leray-Schauder fixed point theorem. All other steps -the derivation of a priori estimates by testing the approximate equation with u, the extraction of weakly convergent subsequences and the proof that the weak limits are identical with their expected limitsare in complete analogy with the proof given in [10] for the prototypical equation (2). Hence, under the structural assumptions (A1)-(A4) there exists a weak solution of (1).…”

Section: Introductionmentioning

confidence: 87%

“…as u Y → ∞, then f (u) automatically satisfies assumptions similar to those of Remark 1.4, so that weak solutions of equation 1exist. Indeed, apply inequality (10)…”

Section: Strong Solutionsmentioning

confidence: 99%

See 2 more Smart Citations

Strong Solutions of Doubly Nonlinear Parabolic Equations

Matas¹,

Merker²

2012

Z. Anal. Anwend.

Self Cite

View full text Add to dashboard Cite

The aim of this article is to discuss strong solutions of doubly nonlinear parabolic equations ∂Bu ∂t + Au = f, where A : X → X * and B : Y → Y * are operators satisfying standard assumptions on boundedness, coercivity and monotonicity. Six different situations are identified which allow to prove the existence of a solution u ∈ L ∞ (0, T ; X ∩ Y) to an initial value u 0 ∈ X ∩ Y , but only in some of these situations the equation is valid in a stronger space than (X ∩ Y) * .

show abstract

Section: Consider the Right Hand Side As An Operatormentioning

confidence: 62%

Section: Introductionmentioning

confidence: 87%

See 1 more Smart Citation

Strong Solutions of Doubly Nonlinear Parabolic Equations

Matas¹,

Merker²

2012

Z. Anal. Anwend.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Combining the result in [26] and the fact that |∇v ǫ k | p−2 ∇v ǫ k ⇀ χ in L q (0, T ; L q (R d )), we have…”

Section: 2mentioning

confidence: 77%

p-Euler equations and p-Navier–Stokes equations

Liu

2018

Journal of Differential Equations

View full text Add to dashboard Cite

Abstract. We propose in this work new systems of equations which we call p-Euler equations and p-Navier-Stokes equations. p-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-p distances, with incompressibility constraint. p-Euler equations have similar structures with the usual Euler equations but the 'momentum' is the signed (p − 1)-th power of the velocity. In the 2D case, the p-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the p-Laplacian of the streamfunction. By adding diffusion presented by γ-Laplacian of the velocity, we obtain what we call p-Navier-Stokes equations. If γ = p, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the p-Navier-Stokes equations in R d for γ = p and p ≥ d ≥ 2 through a compactness criterion.

show abstract

“…Using the regular approximation in Proposition 1 and the arguments of Lemma 2.2 in [15] and Lemma 1.8 in [1], we have the following result.…”

mentioning

confidence: 90%