Existence of Entropy Solutions for Nonsymmetric Fractional Systems

The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L∞. The solutions attained are unique and exclusive. Performance is established by utilizing the maximum principle for certain generalized time and space-fractional diffusion equations. The fractional differential operator is inspected based on the interpretation of the Riemann–Liouville differential operator. Fractional entropy inequalities are imposed.


Introduction
Fractional order differential equations have been positively engaged in modeling of various different procedures and schemes in engineering, physics, chemistry, biology, medicine, and food processing [1][2][3][4].In these requests, reflecting boundary value problems such as the existence and uniqueness of solutions for space-time fractional diffusion equations on bounded domains is a significant procedure.The existence and uniqueness of solutions for linear and nonlinear fractional differential equations has fascinated many investigators [5][6][7][8][9][10][11][12][13].
Fractional calculus created from the Riemann-Liouville description of fractional integral of order ℘ is in the form The fractional order differential of the function φ of order ℘ > 0 is given by When a = 0, we shall denote 0 D ℘ t φ(t) := D ℘ t f (t) and 0 I ℘ t φ(t) := I ℘ t φ(t) in the follow-up.From above, for a = 0, we accomplish that t −℘ , > −1; 0 < ℘ < 1 and t +℘ , > −1; ℘ > 0.
The Leibniz rule for arbitrary differentiations of smooth functions (with continuous derivatives for all orders) φ(t) and ψ(t), t ∈ [a, b] is formulated as (see p. 96 in [14]): where and R ℘ k is the remainder of the series, which can be defined as follows: Additionally, the fractional differential operator achieves linearity (see p. 90 in [14]) Recently, Alsaedi et al. [15] presented an inequality for fractional derivatives related to the Leibniz rule, as follows: Lemma 1.Let one of the following conditions be satisfied Then we have If µ and ν have the same sign and are both increasing or both decreasing, then and for µ = ν, Lemma 1 aims to confirm a conjecture by J. I. Diaz et al. [16].They conjectured that for ℘ ∈ (0, 1), inequality (1) that includes the Riemann-Liouville fractional derivative holds true.We focus on entropy solutions for the fractional Cauchy problem of nonsymmetric systems.We execute sufficient conditions on the parameters to obtain a bounded solutions of L ∞ .The solution is unique and exclusive.Performance is established by applying Lemma 1.The fractional differential operator is inspected according to the interpretation of the Riemann-Liouville differential operator.Various studies have discussed the fractional Cauchy problem [17,18] and entropy analysis [19][20][21].

Proposed Fractional System
We introduce the proposed nonsymmetric fractional system.The Cauchy problem for nonsymmetric system of Keyfitz-Kranzer type is given by the formula [22] The generalization of the system can be written by virtue of the Riemann-Liouville fractional calculus: with bounded measurable initial condition and is a nonlinear function, µ, ω are the density and the velocity of vehicles, while the function Λ is smooth and strictly increasing.The symmetric fractional system of (2) can be viewed as where When n = 1 and Θ(ω) = ω in (4), System (2) reduces to the non symmetric form If we let ν := µω, then we obtain the system with the bounded initial condition For Λ(µ) = µ, system (6) can be viewed as System (2), for an integer case, was addressed by Keyfitz and Krranzer [22] as a model for an elastic string.System (5) was imposed by Aw and Rascle [23] as a macroscopic model for traffic flow, where µ, ω are the density and velocity of vehicles on the road, respectively.Systems ( 6) and ( 7) are pressure-less gas dynamic system models [24].

Solutions and Entropy Solutions
We study the following fractional system based on the above mentioned construction fractional dynamic systems: with the bounded initial condition where t ∈ J := (0, T ], T < ∞, Ω ∈ R 2 is a bounded domain, and the couple denotes the solution of system (8).Moreover, it achieves when µ, ν are smooth in J.
Theorem 1.Let Ω be a bounded domain in R 2 with smooth boundary ∂Ω.Assume that then there exists a unique bounded solution (µ, ν) for system (8).
Proof.The first three steps of the proof describe priori estimates whereas Step 4 addresses uniqueness.
Step 1.First estimate.We aim to prove that (µ, ν) ∈ L 2 (Ω), L 2 (Ω) .By expanding the first equation in ( 8) by µ, utilizing (1) and integrating over Ω, we obtain By applying the Cauchy-Schwartz and Young inequalities, we derive Thus by using the triangle inequality, we obtain Similarly, the product of second equation in ( 8) by ν yields The above equation implies Combining ( 9) and (10) indicates By employing By applying the generalized Gronwall lemma, we achieve where κ 1 , κ 2 and κ 3 are sufficient large positive constants and E ℘ is the Mittag-Leffler function.Hence solution (µ, ν) is bounded in L 2 (Ω).
Step 2. Second estimate.We intend to prove that (µ, ν) Accumulating the first equation in ( 8) by µ (Laplace operator) and integrating over Ω, by considering that µ vanishes on the boundary of Ω Lemma 1, leads to Using this equation, along with the Sobolev embedding, for ∇ν ∈ L 2 (Ω) and ∇Λ ∈ L 2 (Ω) implies that there are two positive constants, namely, K 1 and K 2 such that ∇ν Integration by part for the left hand side of the above inequality, which is based on the Cauchy-Schwartz inequality results in where C 1 := max{K i , i = 1, 2, 3} is a positive constant.Similarly, by multiplying the second equation in ( 8) by ν, and kipping in mind that ν vaporizes on the boundary of Ω, Lemma 1 implies that which, together with the Sobolev embedding, yields positive value of constant 1 satisfying ∇µ L 2 ≤ 1 .Thus, we have where C 2 := max{ 1 , K 2 , K 3 } is a positive constant.Combining ( 13) and ( 14) implies that where C := max{C 1 , C 2 } is a positive constant.By exploiting the generalized Gronwall lemma and the condition Step 3. Upper bound.We aim to determine the upper bound of the fractional derivative.Let Given that µ and ν vanish at the boundary of Ω, we conclude that thus from [15], Remark 2, we have Operating (15) by I α , we derive where Ψ := (τ ) + λ(τ ) + 1. Simple calculation implies Hence, for all t ∈ J, we realize that sup t∈(0,T ] (t) ≤ α, where α is a positive constant depending on ℘, C, 0 and sup t∈J Ψ .
Step 4. Uniqueness.Let (υ 1 , υ 2 ) and (ν 1 , ν 2 ) be two solutions for system (8) under the identical initial condition Multiply the first equation in ( 19) by µ and the second equation in ( 19) by ν and integrate over Ω to obtain relation (12).By employing the generalized Gronwall lemma, we conclude that where σ is an arbitrary constant depending on T, ℘ and the initial condition.System (8) admits a unique bounded global solution (µ, ν) of arbitrary initial value, satisfying µ 2 ≥ ν 2 .This completes the proof.
Subsequently, we discuss the solutions for system (8) when µ 2 ≤ ν 2 .In this case, we only obtain entropy solutions.
Theorem 2. Let Ω be a bounded domain in R 2 with smooth boundary ∂Ω.Assume that (8) satisfies the entropy fractional inequality where Proof.Multiplying the first equation in (8) by ln µ, integrating over Ω and exploiting Lemma 1, we arrive at By utilizing the Cauchy-Schwartz inequality and yielding that µ vanishes on Ω, we take out Thus, by using (see [25]) we obtain Based on our assumption (ν 2 ≥ µ 2 ), we conclude that We arrive at the desired assertion by combining ( 21) and ( 22).This step completes the proof.
Theorem 3. Let Ω be a bounded domain in R 2 with smooth boundary ∂Ω.Assume that If ν 2 ≥ µ 2 then system (8) admits a bounded entropy solution.
It suffices to show that the fractional operator D ℘ t in ( 23) is bounded.Multiplying the first equation in (23) by ln µ, integrating over Ω, exploiting Lemma 1, employing the Cauchy-Schwartz inequality and defining that µ vanishes on Ω, we deduce By considering the earlier observation [25] Ω and since Ω µ ≤ Ω µ 0 , which leads to then the inequality (24) reduces to Similarly, we may infer Combining ( 25) and ( 26) and letting → 0, we arrive at Hence, the proof is completed.
Proof.Again it be adequate to present that the fractional operator D ℘ t in (27) is bounded.Similar to the procedure in Theorem 3, we deduce that by multiplying the first equation in (27) by ln µ, integrating over Ω, exploiting Lemma 1, applying the Cauchy-Schwartz inequality and determining that µ vanishes on Ω, we conclude that Since (see [25]) Hence, the proof is completed.
Corollary 1.Let the hypotheses of Theorem 4 hold.Then for ε → 0, system (8) has a bounded entropy solution.