The cauchy problem and the mixed boundary value problem for a non-linear hyperbolic partial differential equation in two independent variables

1959 ◽  
Vol 3 (1) ◽  
pp. 355-380 ◽  
Author(s):  
James Conlan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Boundary Value Problem ◽  
Mixed Boundary ◽  
Mixed Boundary Value Problem ◽  
Hyperbolic Partial Differential Equation ◽  
Independent Variables ◽  
The Cauchy Problem ◽  
Two Independent Variables

Estimates for Solutions of Wave Equations with Vanishing Curvature

1985 ◽  
Vol 37 (6) ◽  
pp. 1176-1200 ◽  
Author(s):  
Bernard Marshall
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Linear Combination ◽  
Initial Value Problem ◽  
Wave Equations ◽  
Hyperbolic Partial Differential Equation ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Image Position

The solution of the Cauchy problem for a hyperbolic partial differential equation leads to a linear combination of operators Tt of the formFor example, the solution of the initial value problemis given by u(x, t) = Ttf(x) wherePeral proved in [11] that Tt is bounded from LP(Rn) to LP(Rn) if and only ifFrom the homogeneity, the operator norm satisfies ‖Tt‖ ≦ Ct for all t > 0.

scholarly journals Continuity equations and ODE flows with non-smooth velocity

2014 ◽  
Vol 144 (6) ◽  
pp. 1191-1244 ◽  
Author(s):  
Luigi Ambrosio ◽  
Gianluca Crippa
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Bounded Variation ◽  
Velocity Fields ◽  
Well Posedness ◽  
Continuity Equations ◽  
Quantitative Estimates ◽  
The Cauchy Problem

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

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