The Cauchy problem for the three-dimensional Euler-Boltzmann equations of a polytropic, ideal and isentropic fluid in radiation hydrodynamics is considered. The formation of singularities in smooth solutions is studied. It is proved that some C 1 solutions, regardless of the size of the initial disturbance, will develop singularities in a finite time provided that the initial disturbance satisfies certain conditions.
In this paper, we are dealing with an analytical study of a singular fractional order nonlinear differential equation with fractional integral and differential boundary conditions and p -operator, for existence and stability results. Our problem is based on two types of fractional order derivatives, that is, Caputo factional derivative of order and Riemann-Liouville derivative of order , where m − 1 < , ≤ m, and m ∈ {3, 4, 5, … }. The suggested problem will be converted into an equivalent integral form by the help of Green function. After the proofs for these properties, some classical fixed point theorems are employed for the existence of positive solution (EPS). For application of the results, an expressive example is included. KEYWORDS Caputo fractional derivative, existence of positive solution, Hyers-Ulam stability, Riemann-Liouville fractional derivative, singular fractional differential equations c u(t) = u(t) + (t, u(t)), u ′ (t) = 0, u(0) = h(u(T)), where 0 < T < +∞, ∈ R + , , h are given functions and c is Caputo derivative of fractional order ∈ (1, 2). Baleanu et al 15 studied EUS for the following equation: Math Meth Appl Sci. 2018;41:9321-9334.wileyonlinelibrary.com/journal/mma existence results for a class of nonlinear fractional differential equations with singularity. Math Meth Appl Sci. 2018;41:9321-9334.
The Cauchy problem for the one-dimensional Euler-Boltzmann equations in radiation hydrodynamics is studied. The global weak entropy solutions are constructed using the Godunov finite difference scheme. The global existence of weak entropy solutions in L ∞ with arbitrarily large initial data is established with the aid of the compensated compactness method.
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