On Some Interpolation Inequalities Due to Olga Ladyzhenskaya and Nonlinear Partial Differential Equations

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/e0942-9744-3775-a On Some Interpolation Inequalities Due to Olga Ladyzhenskaya and Nonlinear Partial Differential Equations Degtyarev, S. P. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 3. Abstract: We consider some multiplicative interpolation inequalities between the Holder space and the Lebesgue space. Multiplicative interpolation inequalities of the Gagliardo-Nirenberg type are used in the investigations of partial differential equations. Several such inequalities involving the Holder norm (seminorm) were already proved and applied. In the present paper we generalise previous results to the anisotropic "parabolic" case with another simple proof due to idea of Olga Ladyzhenskaya. The manuscript also contains an application of such Gagliardo-Nirenberg type inequality with the Holder norm. Some integral estimate and this inequality give a priori estimate of the solution to quasilinear parabolic problem in the smooth Holder classes. Moreover, using this a priori estimate, we establish the existence of solution of the quasilinear parabolic problem. In order to prove multiplicative inequality of the Gagliardo-Nirenberg type with the Holder norm we use an equivalent normalization of the higher order Holder spaces over higher order finite differences. The key technical tool is the representation of a function \(u(x,t)\) at an arbitrary fixed point \((x,t)\) over a higher order finite difference at this point and the corresponding additional sum of values at neighboring points. After that we integrate with respect to the neighboring points over the balls \(B_{r}((x,t))\) of small radius \(r\). Estimating the finite difference over the corresponding Holder seminorm, we obtain an additive inequality with the parameter \(r\), involving the Holder and integral norms. Optimizing this inequality over \(r\) we get the multiplicative estimate of the Gagliardo-Nirenberg type with the Holder norm and the Lebesgue norm. Keywords: interpolation inequalities, a priori estimates, nonlinear PDE Language: English Download the full text Download JATS XML For citation: Degtyarev, S. P. On Some Interpolation Inequalities Due to Olga Ladyzhenskaya and Nonlinear Partial Differential Equations, Vladikavkaz Math. J., 2025, vol. 27, no. 3, pp.40-49. DOI 10.46698/e0942-9744-3775-a + References 1. Lunardi, A. Interpolation Theory, Lecture Notes (Scuola Normale Superiore), Pisa, Edizioni Della Normale, 2018, 200 p. DOI: 10.1007/978-88-7642-638-4. 2. Lunardi, A. Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, vol. 16, Basel, Birkhauser Verlag, 1995, 424 p. 3. Triebel, H. Theory of Function Spaces II, Reprint of the 1992 Edition, Birkhauser, Basel, Modern Birkhauser Classics, 2010, 370 p. DOI: 10.1007/978-3-0346-0419-2. 4. Kufner, A. and Wannebo, A. An Interpolation Inequality Involving Holder Norms, Georgian Mathematical Journal, 1995, vol. 2, no. 6, pp. 603-612. DOI: 10.1007/BF02262857. 5. Mengxia Dong. Gagliardo-Nirenberg Inequality with Holder Norms, arXiv: 2405.00941, 2024. URL: https://arxiv.org/abs/2405.00941. 6. Triebel, H. Gagliardo-Nirenberg Inequalities, Proceedings of the Steklov Institute of Mathematics, 2014, vol. 284, pp. 263-279. DOI: 10.1134/S0081543814010192. 7. Kolyada, V. I. and Perez Lazaro, F. J. On Gagliardo-Nirenberg Type Inequalities, Journal of Fourier Analysis and Applications, 2014, vol. 20, pp. 577-607. DOI: 10.1007/s00041-014-9320-y. 8. Soudsky, F., Molchanova, A. and Roskovec, T. Interpolation Between Holder and Lebesgue Spaces with Applications, Journal of Mathematical Analysis and Applications, 2018, vol. 466, no. 1, pp. 160-168. DOI: 10.1016/j.jmaa.2018.05.067. 9. Wei, W., Wang, Y. and Ye, Y. Gagliardo-Nirenberg Inequalities in Lorentz Type Spaces, Journal of Fourier Analysis and Applications, 2023, vol. 29, article no. 35, DOI: 10.1007/s00041-023-10018-2. 10. Strzelecki, P. Gagliardo-Nirenberg Inequalities with a BMO Term, Bulletin of the London Mathematical Society, 2006, vol. 38, no. 2, pp. 294-300. DOI: 10.1112/S0024609306018169. 11. Ladyzhenskaya, O. A. A Theorem on Multiplicators in Nonhomogeneous Holder Spaces and Some of Its Applications, Journal of Mathematical Sciences, New York, 2003, vol. 115, no. 6, pp. 2792-2802. DOI: 10.1023/A:1023373920221. 12. Solonnikov, V. A. Estimates for Solutions of a Non-Stationary Linearized System of Navier-Stokes Equations, Boundary Value Problems of Mathematical Physics, Part 1, Collection of Articles, Trudy Matematicheskogo Instituta Imeni V. A. Steklova, Moscow-Leningrad, Nauka, 1964, vol. 70, pp. 213-317 (in Russian). 13. Golovkin, K. K. On Equivalent Normalizations of Fractional Spaces, Automatic Programming, Numerical Methods and Functional Analysis, Trudy Matematicheskogo Instituta Imeni V. A. Steklova, Moscow-Leningrad, 1962, vol. 66, pp. 364-383 (in Russian). 14. Ladyzhenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N. Linear and Quasilinear Equations of Parabolic Type,Translations of Mathematical Monographs, American Mathematical Society, 1968, vol. 23, 648 p. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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