scholarly journals On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

2015 ◽  
Vol 17 (03) ◽  
pp. 1450028 ◽  
Author(s):  
Zhuoping Ruan ◽  
Ingo Witt ◽  
Huicheng Yin
Keyword(s):  
Initial Data ◽  
Local Existence ◽  
Piecewise Smooth ◽  
Degree Zero ◽  
Essential Difference ◽  
Existence Of A Solution ◽  
Discontinuous Initial Data ◽  
Tricomi Equation ◽  
Low Regularity Solution ◽  
Straight Lines

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation [Formula: see text] with typical discontinuous initial data (u(0, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x1 = 0}, it is shown that the local solution u(t, x) ∈ L∞([0, T] × ℝn) exists and is C∞ away from the forward cuspidal conic surface [Formula: see text] or the cuspidal wedge-shaped surfaces [Formula: see text] respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x1 = 0} and {t = x2 = 0}, we establish the local existence of a solution [Formula: see text] and further show that [Formula: see text] in general due to the degenerate character of the equation under study, where [Formula: see text]. This is an essential difference to the well-known result for solution [Formula: see text] to the two-dimensional semilinear wave equation [Formula: see text] with (v(0, x), ∂tv(0, x)) = (0, φ(x)), where Σ0 = {t = |x|}, [Formula: see text] and [Formula: see text].

Local existence with low regularity for the 2D compressible Euler equations

2021 ◽  
Vol 18 (03) ◽  
pp. 701-728
Author(s):  
Huali Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Euler Equations ◽  
Local Existence ◽  
Compressible Euler Equations ◽  
Two Dimensional ◽  
Spatial Derivative ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Local Solutions

We prove the local existence, uniqueness and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, where the initial data of velocity, density, specific vorticity [Formula: see text] and the spatial derivative of specific vorticity [Formula: see text].

scholarly journals Well posedness for the Kawahara equation on the half-line

2020 ◽  
Author(s):  
Boling Guo ◽  
fengxia liu
Keyword(s):  
Initial Data ◽  
Local Existence ◽  
Half Line ◽  
Well Posedness ◽  
Kawahara Equation ◽  
Low Regularity ◽  
Regularity Properties

We study the low-regularity properties of the Kawahara equation on the half line. We obtain the local existence, uniqueness, and continuity of the solution. Moreover, We obtain that the nonlinear terms of the solution are smoother than the initial data.

scholarly journals On the proximity of solutions of unperturbed and hyperbolized heat equations for discontinuous initial data

2017 ◽  
pp. 1-15
Author(s):  
Tikhon Evgenievich Moiseev ◽  
Elena Evgenievna Myshetskaya ◽  
Vladimir Fedorovich Tishkin
Keyword(s):  
Initial Data ◽  
Heat Equations ◽  
Discontinuous Initial Data

scholarly journals Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Lihua Deng ◽  
Xianguang Shang
Keyword(s):  
Parabolic Equation ◽  
Initial Data ◽  
Degenerate Parabolic Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Local Existence ◽  
Time Dependent ◽  
Priori Estimates ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

This paper is devoted to the Cauchy problem for a class of doubly degenerate parabolic equation with time-dependent gradient source, where the initial data are Radon measures. Using the delicate a priori estimates, we first establish two local existence results. Furthermore, we show that the existence of solutions is optimal in the class considered here.

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.

PARAXIAL APPROXIMATION OF THE VLASOV-MAXWELL SYSTEM: LAMINAR BEAMS

1994 ◽  
Vol 04 (02) ◽  
pp. 203-221 ◽  
Author(s):  
A. NOURI
Keyword(s):  
Electromagnetic Field ◽  
Charged Particle ◽  
Initial Data ◽  
Global Existence ◽  
Local Existence ◽  
Sufficient Conditions ◽  
External Electromagnetic Field ◽  
Evolution System ◽  
Lagrangian Coordinates ◽  
Maxwell System

The Vlasov-Maxwell stationary system for charged particle laminar beams is studied with a paraxial model of approximation. It leads to a degenerate evolution system, which local existence is proved. Then, using lagrangian coordinates, with sufficient conditions on the initial data and the external electromagnetic field, it is shown that global existence is possible.

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