smooth solutions
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Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 15
Ernesto Guerrero Fernández ◽  
Cipriano Escalante ◽  
Manuel J. Castro Díaz

This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

Sayed Arsalan Sajjadi ◽  
Hashem Saberi Najafi ◽  
Hossein Aminikhah

The solutions of weakly singular fractional integro-differential equations involving the Caputo derivative have singularity at the lower bound of the domain of integration. In this paper, we design an algorithm to prevail on this non-smooth behaviour of solutions of the nonlinear fractional integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated. The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.

Zhonger Wu ◽  
Zhong Tan ◽  

In this paper, we consider the Cauchy problem of the compressible quantum Navier-Stokes equations with damping in R3. We first assume that the H3-norm of the initial data is sufficiently small while the higher derivative can be arbitrarily large, and prove the global existence of smooth solutions. Then the decay estimate of the solution is derived for the initial data in a homogeneous Sobolev space or Besov space with negative exponent. In addition, the usual Lp−L2(1 ≤ p ≤ 2) type decay rate is obtained without assuming that the Lpnorm of the initial data is sufficiently small.

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 84-109
Sandeep Kumar

Abstract The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the Euclidean space, a nonzero torsion now implies two different helical curves. This generalizes recent works by the author with de la Hoz and Vega on helical polygons in the Euclidean space as well as planar polygons in the Minkowski space. Numerical experiments in this article show that the trajectory of the point X(0, t) exhibits new variants of Riemann’s non-differentiable function whose structure depends on the initial torsion in the problem. As a result, we observe that the smooth solutions (helices, straight line) in the Minkowski space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega.

2021 ◽  
Vol 26 (4) ◽  
pp. 519-527
Jishan Fan ◽  
Peng Wang ◽  
Yong Zhou

In this paper, we are concerned with the uniform regularity estimates of smooth solutions to the isentropic compressible magneto-micropolar system in T3. Under the assumption that , and by applying the classic bilinear commutator and product estimates, the uniform estimates of solutions to the isentropic compressible magneto-micropolar system are established in space, .

Anna Fino ◽  
Gueo Grantcharov ◽  
Luigi Vezzoni

AbstractWe construct new smooth solutions to the Hull–Strominger system, showing that the Fu–Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $$13 \le k \le 22$$ 13 ≤ k ≤ 22 and $$14\le r\le 22$$ 14 ≤ r ≤ 22 , the smooth manifolds $$S^1\times \sharp _k(S^2\times S^3)$$ S 1 × ♯ k ( S 2 × S 3 ) and $$\sharp _r (S^2 \times S^4) \sharp _{r+1} (S^3 \times S^3)$$ ♯ r ( S 2 × S 4 ) ♯ r + 1 ( S 3 × S 3 ) , have a complex structure with trivial canonical bundle and admit a solution to the Hull–Strominger system.

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