For the physical vacuum free boundary problem with the sound speed being C 1/2 - H¨older continuous near vacuum boundaries of the compressible Euler equations with damping, the global existence of solutions and convergence to Barenblatt self-similar solutions of the porous media equation was recently proved in  for 1-d motions by Luo and the author. This paper generalizes the results for 1-d motions to 3-d spherically symmetric motions. Compared with the 1-d theory, besides the high degeneracy of the equations near the physical vacuum boundary, the analytical difficulties lie in the complexity of equations and the coordinates singularity in the center of symmetry which is resolved by constructing suitable weights. The results obtained in this work contribute to the theory of global solutions to free boundary problems of compressible inviscid fluids in the presence of vacuum states, for which the currently available results are mainly for the local-in-time well-posedness theory, also to the theory of global smooth solutions of dissipative hyperbolic systems which fail to be strictly hyperbolic.
We provide a unified approach for constructing Wick words in mixed q-Gaussian algebras, which are generated by sj = aj +a ∗ j , j = 1, · · · , N, where aia ∗ j −qija ∗ j ai = δij . Here we also allow equality in −1 ≤ qij = qji ≤ 1. This approach relies on Speicher’s central limit theorem and the ultraproduct of von Neumann algebras. We also use the unified argument to show that the Ornstein–Uhlenbeck semigroup is hypercontractive, the Riesz transform associated to the number operator is bounded, and the number operator satisfies the Lp Poincar´e inequalities with constants C √p. Finally we prove that the mixed q-Gaussian algebra is weakly amenable and strongly solid in the sense of Ozawa and Popa. Our approach is mainly combinatorial and probabilistic. The results in this paper can be regarded as generalizations of previous results due to Speicher, Biane, Lust-Piquard, Avsec, et al.
We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps satisfy certain "bounded measure type condition", which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by Lau, Wang and Chu.
For a self-similar measure in d-dimensional Euclidean space with overlaps but satisfies the so-called bounded measure type condition introduced by Tang and the authors, we set up a framework for deriving a closed formula for the Lq-spectrum of the measure for nonnegative q. The framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the Lq-spectrum have only been obtained earlier for measures satisfying Strichartz second-order identities. We illustrate how to use our results to prove the differentiability of the Lq-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.
We consider the mixed q-Gaussian algebras introduced by Speicher which are generated by the variables Xi = li + l ∗ i , i = 1, . . . , N, where l ∗ i lj − qij lj l ∗ i = δi,j and −1 < qij = qji < 1. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed q-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra L(FN ), provided that maxi,j |qij | is small enough. Similar results hold in the reduced C ∗ -algebra setting. The proof relies on some estimates which are generalizations of Dabrowski’s results for the special case qij ≡ q.