scholarly journals Existence and Uniqueness of Weak Solutions to Viscous Primitive Equations for a Certain Class of Discontinuous Initial Data

2017 ◽  
Vol 49 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Jinkai Li ◽  
Edriss S. Titi
Keyword(s):  
Initial Data ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Primitive Equations ◽  
Discontinuous Initial Data

scholarly journals Delay differential systems with discontinuous initial data and existence and uniqueness theorems for systems with impulse and delay

1994 ◽  
Vol 7 (1) ◽  
pp. 49-67 ◽  
Author(s):  
S. V. Krishna ◽  
A. V. Anokhin
Keyword(s):  
Initial Data ◽  
Existence And Uniqueness ◽  
Fixed Time ◽  
Impulsive System ◽  
Number Of Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Discontinuous Initial Data ◽  
Delay Differential Systems ◽  
Delay Differential

The main purpose of this paper is to discuss some qualitative aspects of differential equations with delays and impulses. Such systems are encountered in modeling the dynamics of prices and cultured populations. However, any such discussion has to be based on some existence and uniqueness results for delay equations with discontinuous initial data. This is the content of the first part of the paper. For an impulsive system, we observe a phenomenon of existence of infinite number of solutions subject to impulses arbitrarily close to a fixed time. Conditions, when such solutions exist and when they do not, are discussed.

scholarly journals Global Weak Solutions for the Weakly Dissipative Dullin-Gottwald-Holm Equation

2021 ◽  
pp. 91-108
Author(s):  
Shiyu Li
Keyword(s):  
Initial Data ◽  
Surface Waves ◽  
Shallow Water ◽  
Strong Solution ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Water Regime ◽  
Global Weak Solutions ◽  
Holm Equation ◽  
Sign Conditions

In this paper, we are concerned with the existence and uniqueness of global weak solutions for the weakly dissipative Dullin-Gottwald-Holm equation describing the unidirectional propagation of surface waves in shallow water regime:                                        ut − α2uxxt + c0ux + 3uux + γuxxx + λ(u − α2uxx) = α2(2uxuxx + uuxxx).Our main conclusion is that on c0 = − γ/α2 and λ ≥ 0, if the initial data satisfies certain sign conditions, then we show that the equation has corresponding strong solution which exists globally in time, finally we demonstrate the existence and uniqueness of global weak solutions to the equation.

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.

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