论文标题
非局部速度的保护法 - 奇异极限问题
Conservation laws with nonlocal velocity -- the singular limit problem
论文作者
论文摘要
我们考虑具有非局部速度的保护定律，并显示了指数类型的非局部权重，当非局部保护法接近dirac分布时，独特的解决方案以弱或强的意义（取决于速度的规律性）在弱或强的意义上（取决于速度的规律性）。为此，我们首先在非局部速度上建立了均匀的总变化估计，这使其能够证明非局部溶液在极限中是可允许的熵。对于熵解决方案，我们使用量身定制的熵通量对，该对仅允许使用一个熵来获得独特性（给定一些其他约束）。 对于一般的权重，我们表明随着时间的推移，保留了初始基准的单调性，这使其能够证明与局部熵解决方案的收敛性，也可以证明相当一般的内核和单调初始基准。这涵盖了当地保护法的原型：冲击波和稀有因素。它还强调说，``速度中的非本地''近似可能更适合于近似局部保护定律，而不是在解决方案近似中的非局部性，而这种单调性仅适用于特定速度。
We consider conservation laws with nonlocal velocity and show for nonlocal weights of exponential type that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we establish first a uniform total variation estimate on the nonlocal velocity which enables it to prove that the nonlocal solution is entropy admissible in the limit. For the entropy solution, we use a tailored entropy flux pair which allows the usage of only one entropy to obtain uniqueness (given some additional constraints). For general weights, we show that monotonicity of the initial datum is preserved over time which enables it to prove the convergence to the local entropy solution for rather general kernels and monotone initial datum as well. This covers the archetypes of local conservation laws: Shock waves and rarefactions. It also underlines that a ``nonlocal in the velocity'' approximation might be better suited to approximate local conservation laws than a nonlocal in the solution approximation where such monotonicity does only hold for specific velocities.
