Estimates for fundamental solutions of parabolic equations in non-divergence form

Dong, Hongjie; Kim, Seick; Lee, Sung‐Jin

doi:10.1016/j.jde.2022.09.007

Cited by 4 publications

(8 citation statements)

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“…In other words, we need to devise appropriate weighted topologies in which the gluing system becomes weakly coupled and thus can be solved by fixed point arguments. For the outer problem, we make use of the sub-Guassian estimate recently proved in [20]. For the inner problem, good solutions with sufficient decay in space and time can only be captured with careful choices of the parameters λ j , γ j , ξ [j] .…”

Section: • Ansatz Of Multiple Bubblesmentioning

confidence: 99%

“…The linear theory for the outer problem is done by using the sub-Gaussian estimate for the linearized parabolic system proved in [20], where DMO x -regularity is required for the entries in the matrix B Φ,U * . The dependence on Φ in the matrix shall be dealt with via Schauder fixed point theorem, and the key thing here is the dependence on the blow-up profile U * .…”

Section: • Non-local Reduced Problemsmentioning

confidence: 99%

“…for all (t, x) ∈ R d+1 , ξ ∈ R d , and θ ∈ R n . We use the symbol in [20] to give some basis definitions. Define the parabolic distance between X = (t, x) and…”

Section: Orthogonal Equationsmentioning

confidence: 99%

“…Assume the principal coefficients A αβ ∈ DMO x (R n+1 ). Then Theorem 1.3 in [20] can be generalized to parabolic systems (7.1) (see [20]). 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.…”

Section: Orthogonal Equationsmentioning

confidence: 99%

“…So one cannot expect good Hölder continuity in the coefficients and has to work in certain weaker class. 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 (DMO x ), which was proved by Dong, Kim and Lee [20] recently. We introduce Dini mean absolute oscillation in space (|DMO| x ), which is a subspace of DMO x .…”

Section: Introductionmentioning

confidence: 99%

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

Wei¹,

Zhang²,

Zhou³

2022

Preprint

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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

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Section: • Ansatz Of Multiple Bubblesmentioning

confidence: 99%

Section: • Non-local Reduced Problemsmentioning

confidence: 99%

“…for all (t, x) ∈ R d+1 , ξ ∈ R d , and θ ∈ R n . We use the symbol in [20] to give some basis definitions. Define the parabolic distance between X = (t, x) and…”

Section: Orthogonal Equationsmentioning

confidence: 99%

Section: Orthogonal Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Wei¹,

Zhang²,

Zhou³

2022

Preprint

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Green’s Function of the Cauchy Problem for Equations with Dissipative Parabolicity, Negative Genus, and Variable Coefficients

Litovchenko,

Kharyna

2024

International Journal of Differential Equations

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Green’s function of the Cauchy problem is constructed by the method of successive approximations, and its main properties are studied for a new class of linear differential equations with dissipative parabolicity and negative genus, whose coefficients are bounded, continuous in time, and infinitely differentiable by the spatial variable of the function. This class covers Shilov parabolic equations, as well as the class of the Zhitomirskii parabolic Shilov‐type equations with variable coefficients and negative genus, and successfully complements the Petrovsky class of parabolic equations.

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Holder Solvability of Quasi-Linear Parabolic Systems in the Divergent Form

Yaremenko

2024

Pan-Amer. J. Math.

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Estimates for fundamental solutions of parabolic equations in non-divergence form

Cited by 4 publications

References 10 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

Green’s Function of the Cauchy Problem for Equations with Dissipative Parabolicity, Negative Genus, and Variable Coefficients

Holder Solvability of Quasi-Linear Parabolic Systems in the Divergent Form

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