The simplest quasi-linear equation is ut + a(f u))x = 0, where a is a given discontinuous coefficient function and f is a smooth function.
If u is a smooth divergence-free vector field of (mathbb {R}^{2}) with vorticity, and f is a smooth function, then for all (pin 1, infty)), begin{aligned} biglVert [mathcal{R},ucdotnabla]fbigrVert _{L^{p}}leq CVert nabla uVert _{L^{p}}bigl Vert f Vert _{L^{2}}+Vert fVert _{B^{0}_{{infty},1}}bigr). end{aligned} (2.7).
Let f be a smooth function with compact support.
There is a useful remark here: if f is a real-valued smooth function on the sphere, its derivative df provides us with an element of (varOmega^{1}_{mathbb{L}^{2}}(mathbb {S}^{2})).
Lemma 4.1 Assume that f = f ( v ) is a smooth function.
If f is a function having an inverse f −1 (a function that "undoes" the original function), the number is called the mean value of x1, x2, …, xn associated with f.
holds if and only if f is a linear function.
Theorem 3.1 Let s ≥ 0. Suppose that f ( v ) is a smooth function and f ′ ( 0 ) > − α.
If f is a log-convex function then f is also a convex function.
Here a vector valued function f ( t ) is called piecewise smooth on [ 0, ∞ ), if there exist 0 < t 1 < t 2 < ⋯ < t k < + ∞ such that f ( t ) is a smooth function on ( 0, t 1 ), ( t i, t i + 1 ), i = 1, 2, …, k − 1 and ( t k, ∞ ), respectively.
