article published yes Giovanni Brigati author 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1 Jean Dolbeault author Nikita Simonov author JaMa department IST-BRIDGE: International postdoctoral program project In this paper, we review recent results on stability and instability in logarithmic Sobolev inequalities, with a particular emphasis on strong norms. We consider several versions of these inequalities on the Euclidean space, for the Lebesgue and the Gaussian measures, and discuss their differences in terms of moments and stability. We give new and direct proofs, as well as examples and discuss the stability of a logarithmic uncertainty principle. Although we do not cover all aspects of the topic, we hope to contribute to establishing the state of the art. https://research-explorer.ista.ac.at/download/21018/21025/2026_LaMatematica_Brigati.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Springer Nature2026 eng 2730-9657 2504.0865810.1007/s44007-025-00180-y 5 Brigati G, Dolbeault J, Simonov N. Logarithmic Sobolev Inequalities: A review on stability and instability results. <i>La Matematica</i>. 2026;5. doi:<a href="https://doi.org/10.1007/s44007-025-00180-y">10.1007/s44007-025-00180-y</a> Brigati, Giovanni, et al. “Logarithmic Sobolev Inequalities: A Review on Stability and Instability Results.” <i>La Matematica</i>, vol. 5, 5, Springer Nature, 2026, doi:<a href="https://doi.org/10.1007/s44007-025-00180-y">10.1007/s44007-025-00180-y</a>. Brigati, Giovanni, Jean Dolbeault, and Nikita Simonov. “Logarithmic Sobolev Inequalities: A Review on Stability and Instability Results.” <i>La Matematica</i>. Springer Nature, 2026. <a href="https://doi.org/10.1007/s44007-025-00180-y">https://doi.org/10.1007/s44007-025-00180-y</a>. Brigati G, Dolbeault J, Simonov N. 2026. Logarithmic Sobolev Inequalities: A review on stability and instability results. La Matematica. 5, 5. G. Brigati, J. Dolbeault, N. Simonov, La Matematica 5 (2026). Brigati, G., Dolbeault, J., &#38; Simonov, N. (2026). Logarithmic Sobolev Inequalities: A review on stability and instability results. <i>La Matematica</i>. Springer Nature. <a href="https://doi.org/10.1007/s44007-025-00180-y">https://doi.org/10.1007/s44007-025-00180-y</a> G. Brigati, J. Dolbeault, and N. Simonov, “Logarithmic Sobolev Inequalities: A review on stability and instability results,” <i>La Matematica</i>, vol. 5. Springer Nature, 2026. 210182026-01-20T10:14:55Z2026-01-21T07:48:28Z