Sobolev inequality and its applications to nonlinear PDE on noncommutative Euclidean spaces
Ruzhansky M. Shaimardan S. Tulenov K.
1 December 2025Academic Press Inc.
Journal of Functional Analysis
2025#289Issue 11
In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo–Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the noncommutative Euclidean space. Moreover, we show that the logarithmic Sobolev inequality is equivalent to the Nash inequality for possibly different constants in this noncommutative setting by completing the list in noncommutative Varopouloss theorem in [54]. Finally, we present a direct application of the Nash inequality to compute the time decay for solutions of the heat equation in the noncommutative setting.
Gagliardo–Nirenberg inequality , Noncommutative Euclidean space , Nonlinear damped wave equation , Sobolev inequality
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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom
Institute of Mathematics and Mathematical Modeling, Almaty, 050010, Kazakhstan
Department of Mathematics: Analysis
School of Mathematical Sciences
Institute of Mathematics and Mathematical Modeling
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