On $$\varvec{C^{1/2,1}}$$, $$\varvec{C^{1,2}}$$, and $$\varvec{C^{0,0}}$$ estimates for linear parabolic operators

Dong, Hongjie; Escauriaza, Luis; Kim, Seick

doi:10.1007/s00028-021-00729-8

Cited by 9 publications

(4 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indeed, W 2,p estimate for parabolic systems is given in [18], which can be used to generalize Lemma 2.2 in [20] to parabolic systems. Lemma 2.3 in [20], which is first given in Theorem 3.3 in [17] can be also generalized to parabolic systems.…”

Section: Orthogonal Equationsmentioning

confidence: 94%

“…where α is an arbitrary number in (0, 1). [17] and generalized version of Theorem 1.3 in [20], there exists a constant C = C(d, λ, Λ, ω x A , δ) and a universal constant c > 0 such that for 0…”

Section: Orthogonal Equationsmentioning

confidence: 99%

“…The regularity of parabolic system with DMO x coefficients are guaranteed by [17,Theorem 1.2]. Then Schauder fixed point theorem implies the existence of (5.18).…”

Section: Solving the Gluing Systemmentioning

confidence: 99%

See 2 more Smart Citations

Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two

Wei¹,

Zhang²,

Zhou³

2022

Preprint

View full text Add to dashboard Cite

We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from R 2 into S 2 ut = a(∆u + |∇u| 2 u) − bu ∧ ∆u in R 2 × (0, T ),whereGiven any prescribed N points in R 2 and small T > 0, we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time t = T , taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speedThe proof is based on the parabolic inner-outer gluing method, developed in [12] for Harmonic Map Flow (HMF). However, substantial difficulties arise due to the coupling between HMF and Schrodinger Map Flow (SMF) in LLG, and such coupling produces both dissipative (a > 0) and dispersive (b = 0) features. A direct consequence of the presence of dispersion is the lack of maximum principle for suitable quantities, which makes the analysis more delicate even at the linearized level. The dispersion cannot be treated perturbatively even in the dissipation-dominating case a/|b| >> 1, and one has to include this as part of the leading order. To overcome these difficulties, we make use of two key technical ingredients: first, for the inner problem we employ the tool of distorted Fourier transform, as developed by Krieger, Miao, Schlag and Tataru [34,35]. Second, the linear theory for the outer problem is achieved by means of the sub-Gaussian estimate for the fundamental solution of parabolic system in non-divergence form with coefficients of Dini mean oscillation in space (DMOx), which was proved by Dong, Kim and Lee [20] recently. J. WEI, Q. ZHANG, AND Y. ZHOU 9.3. Higher modes |k| ≥ 2 71 9.4. Mode 0 77 9.5. Mode 1 81 9.6. Mode −1 86 Appendix A. Some useful estimates 95 Appendix B. Convolution estimates in finite time 101 B.1. Preliminary 101 B.2. Convolution about v(t)|x − q| −b 1 {l1(t)≤|x−q|≤l2(t)} 110 B.3. Convolution about v(t)|x − q| −b 1 {|x−q|≥(T −t) 1 2 } 120 Appendix C. Derivation of the weighted topology for the outer problem 123 Appendix D. Estimates of G and H j 129 D.1. Estimates for terms involving Φ out , Φ [j] in 129 D.2. Estimate of ∇ x A 132 D.3. Estimate of G 136 D.4. Estimate of H [j] 161 Acknowledgements 166 References 166

show abstract

Section: Orthogonal Equationsmentioning

confidence: 94%

Section: Orthogonal Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two

Wei¹,

Zhang²,

Zhou³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…(For such an example, of course, we replace the ball |x| < 4 with a smaller one, such as |x| < 1/2.) The paper [14] has spawned a series of applications to various elliptic and parabolic equations: [12], [13], [15]. As useful as the concept of Dini mean oscillation has proven to be, however, we will see that there are equations with coefficients that do not have Dini mean oscillation, yet Lipschitz regularity can be obtained by other methods.…”

Section: Introduction and General Theorymentioning

confidence: 99%

Gilbarg-Serrin Equation and Lipschitz Regularity

Maz′ya¹,

McOwen²

2021

Preprint

View full text Add to dashboard Cite

We discuss conditions for Lipschitz regularity for a uniformly elliptic equation in divergence form with coefficients that were introduced by Gilbarg & Serrin. In particular, we find cases where Lipschitz regularity holds but the coefficients are not Dini continuous, or do not even have Dini mean oscillation. The form of the coefficients also enables us to obtain specific conditions and examples for which there exists a weak solution that is not Lipschitz continuous. MSC: 35B40,

show abstract

Gilbarg-Serrin equation and Lipschitz regularity

Maz′ya

McOwen

2022

Journal of Differential Equations

View full text Add to dashboard Cite

On $$\varvec{C^{1/2,1}}$$, $$\varvec{C^{1,2}}$$, and $$\varvec{C^{0,0}}$$ estimates for linear parabolic operators

Cited by 9 publications

References 38 publications

Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two

Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two

Gilbarg-Serrin Equation and Lipschitz Regularity

Gilbarg-Serrin equation and Lipschitz regularity

Contact Info

Product

Resources

About