Oleı̆nik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term $W:=\mathbb{1}_{(-\infty,0]}(\cdot)\exp(\cdot) \ast \rho$ satisfy an Oleı̆nik-type entropy condition. More precisely, under different sets of assumptions on the velocity function $V$, we prove that $W$ satisfies a one-sided Lipschitz condition and that $V'(W) W \partial_x W$ satisfies a one-sided bound, respectively. As a byproduct, we deduce that, as the exponential kernel is rescaled to converge to a Dirac delta distribution, the weak solution of the nonlocal problem converges to the unique entropy-admissible solution of the corresponding local conservation law, under the only assumption that the initial datum is essentially bounded and not necessarily of bounded variation

A general result on the approximation of local conservation laws by nonlocal conservation laws: The singular limit problem for exponential kernels

13 pages, 2 figuresWe deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables us to obtain a total variation bound on the nonlocal term. By using this, we prove that the (unique) weak solution of the nonlocal problem converges strongly in $C(L^{1}_{\text{loc}})$ to the entropy solution of the local conservation law. We conclude with several numerical illustrations which underline the main results and, in particular, the difference between the solution and the nonlocal term

Protocol for a randomized crossover trial to evaluate the effect of soft brace and rigid orthosis on performance and readiness to return to sport six months post-ACL-reconstruction

A randomized crossover trial was designed to investigate the influence of muscle activation and strength on functional stability/control of the knee joint, to determine whether bilateral imbalances still occur six months after successful anterior cruciate ligament reconstruction (ACLR), and to analyze whether the use of orthotic devices changes the activity onset of these muscles. Furthermore, conclusions on the feedforward and feedback mechanisms are highlighted. Therefore, twenty-eight patients will take part in a modified Back in Action (BIA) test battery at an average of six months after a primary unilateral ACLR, which used an autologous ipsilateral semitendinosus tendon graft. This includes double-leg and single-leg stability tests, double-leg and single-leg countermovement jumps, double-leg and single-leg drop jumps, a speedy jump test, and a quick feet test. During the tests, gluteus medius and semitendinosus muscle activity are analyzed using surface electromyography (sEMG). Motion analysis is conducted using Microsoft Azure DK and 3D force plates. The tests are performed while wearing knee rigid orthosis, soft brace, and with no aid, in random order. Additionally, the range of hip and knee motion and hip abductor muscle strength under isometric conditions are measured. Furthermore, patient-rated outcomes will be assessed