We study a nonlocal regularisation of a scalar conservation law given by a fractional derivative of order between one and two. The nonlocal operator is of Riesz–Feller type with skewness two minus its order. This equation describes the internal structure of hydraulic jumps in a shallow water model. The main purpose of the paper is the study of the vanishing viscosity limit of the Cauchy problem for this equation. First, we study the properties of the solution of the regularised problem and then we show that the difference between the regularised solution and the entropy solution of the scalar conservation law converges to zero in this limit in $$C([0,T];L^1_{loc}({\mathbb {R}}))$$ C ( [ 0 , T ] ; L loc 1 ( R ) ) for initial data in $$L^\infty ({\mathbb {R}})$$ L ∞ ( R ) , and in $$C([0,T];L^1({\mathbb {R}}))$$ C ( [ 0 , T ] ; L 1 ( R ) ) for initial data in $$ L^\infty ({\mathbb {R}})\cap BV({\mathbb {R}})$$ L ∞ ( R ) ∩ B V ( R ) . In order to prove these results we use weak entropy inequalities and the double scale technique of Kruzhkov. Such techniques also allow to show the $$L^1({\mathbb {R}})$$ L 1 ( R ) contraction of the regularised problem. For completeness, we study the behaviour in the tail of travelling wave solutions for genuinely nonlinear fluxes. These waves converge to shock waves in the vanishing viscosity limit, but decay algebraically as $$x-ct \rightarrow \infty $$ x - c t → ∞ , rather than exponentially, the latter being a behaviour that they exhibit as $$x-ct \rightarrow - \infty $$ x - c t → - ∞ , however. Finally, we generalise the results concerning the vanishing viscosity limit to Riesz–Feller operators.

中文翻译：

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由 Riesz-Feller 算子正则化的守恒定律的消失粘度极限

我们研究了由一到二阶的分数阶导数给出的标量守恒定律的非局部正则化。非局部算子属于 Riesz-Feller 类型，偏度为 2 减去其阶数。该方程描述了浅水模型中水跃的内部结构。该论文的主要目的是研究该方程的柯西问题的消失粘度极限。首先，我们研究了正则化问题的解的性质，然后我们证明了正则化解与标量守恒定律的熵解之间的差异在 $$C([0,T]; L^1_{loc}({\mathbb {R}}))$$ C ( [ 0 , T ] ; L loc 1 ( R ) ) 用于 $$L^\infty ({\mathbb {R} })$$ L ∞ ( R ) ，并在 $$C([0,T];L^1({\mathbb {R}}))$$ C ( [ 0 , T ] ; L 1 ( R ) ) 用于 $$ L^\infty ({\mathbb {R}})\cap BV({\mathbb {R}})$$ L ∞ ( R ) ∩ BV ( R ) 中的初始数据。为了证明这些结果，我们使用弱熵不等式和 Kruzhkov 的双尺度技术。这种技术还允许显示正则化问题的 $$L^1({\mathbb {R}})$$ L 1 ( R ) 收缩。为了完整起见，我们研究了真正非线性通量的行波解的尾部行为。这些波在粘度消失极限收敛为冲击波，但代数衰减为 $$x-ct \rightarrow \infty $$ x - ct → ∞ ，而不是指数衰减，后者是它们表现出的行为 $$x- ct \rightarrow - \infty $$ x - ct → - ∞ 然而。最后，我们将有关消失粘度极限的结果推广到 Riesz-Feller 算子。