scholarly journals Continuous maximal regularity on singular manifolds and its applications

2016 ◽  
Vol 5 (2) ◽  
pp. 303-335 ◽  
Author(s):  
Yuanzhen Shao
Keyword(s):  
Maximal Regularity ◽  
Singular Manifolds

scholarly journals $L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Acoustic Waves ◽  
Maximal Regularity ◽  
Bubble Radius ◽  
Fourier Multipliers ◽  
Cylindrical Domain ◽  
Lebesgue Spaces ◽  
Regularity Problem ◽  
Bubbly Liquids

Abstract We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

scholarly journals Lorentz spaces in action on pressureless systems arising from models of collective behavior

2021 ◽  
Author(s):  
Raphaël Danchin ◽  
Piotr Bogusław Mucha ◽  
Patrick Tolksdorf
Keyword(s):  
Initial Data ◽  
Collective Behavior ◽  
Existence And Uniqueness ◽  
Maximal Regularity ◽  
Parabolic Systems ◽  
Lorentz Spaces ◽  
Regularity Estimate ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Time Existence

AbstractWe are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $$L_{q,r}(0,T;L_p(\Omega ))$$ L q , r ( 0 , T ; L p ( Ω ) ) spaces.

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

scholarly journals On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space

2016 ◽  
Vol 15 (05) ◽  
pp. 731-770 ◽  
Author(s):  
D. P. Hewett ◽  
A. Moiola
Keyword(s):  
Maximal Regularity ◽  
Cantor Sets ◽  
Characteristic Functions ◽  
Bessel Potential ◽  
Empty Interior ◽  
Cartesian Products ◽  
Sobolev Regularity ◽  
Full Classification ◽  
Differential And Integral Equations

This paper concerns the following question: given a subset [Formula: see text] of [Formula: see text] with empty interior and an integrability parameter [Formula: see text], what is the maximal regularity [Formula: see text] for which there exists a non-zero distribution in the Bessel potential Sobolev space [Formula: see text] that is supported in [Formula: see text]? For sets of zero Lebesgue measure, we apply well-known results on set capacities from potential theory to characterize the maximal regularity in terms of the Hausdorff dimension of [Formula: see text], sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of [Formula: see text], together with the sets of values of [Formula: see text] for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterizing the regularity that can be achieved on certain special classes of sets, such as [Formula: see text]-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.

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