Cm Math Continuum — Examples

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the actual Lebesgue transport and change-of-variables suppliers.

Let $B:=B_1(0)\subseteq {\mathbb{R}}^3$ be the open unit ball (Open ball, closed ball and sphere in a metric space), a nonempty bounded open set and hence a material region in the sense of Material regions, densities and their mass; let $v\in {\mathbb{R}}^3$ be fixed, and let

$\psi (t,X):=X+tv(t\in \mathbb{R},X\in {\mathbb{R}}^3)$

be the uniform translation motion of Continuum motions, spatial velocity, material acceleration and swept regions (each ${\psi }_t$ is the identity plus a translation, a $C^{\mathrm{\infty }}$ diffeomorphism with $D_X{\psi }_t=I_3$). Let $h:{\mathbb{R}}^3\to \mathbb{R}$ be $C^1$ ($C^k$ Euclidean maps and diffeomorphisms) and define the travelling profile

$f(t,x):=h(x-tv)((t,x)\in \mathbb{R}\times {\mathbb{R}}^3)\mathrm{.}$

Then ${\int }_{{\psi }_t(B)}f(t,x)dx$ is independent of $t$, while the transport integrand vanishes pointwise:

${\mathrm{\partial }}_{t}f+\mathrm{div}(fv)=0,$

so the transport theorem reproduces the constant rate $0$.

• This example verifies the transport identity on a flow with constant velocity and non-constant density-like profile; it exercises the change-of-variables and material-derivative pieces without any physical claim.

Take $\lambda =2$, $\mu =1$, ${\rho }_0=1$ (The small-strain isotropic linear elastic constitutive law and its strain energy: here $\lambda +\frac{2\mu }{3}=\frac{8}{3}>0$). Let $k=(\pi ,0,0)$, $a=(1,0,0)$ and

$\xi (t,x)=a\cos (\pi x_1-\omega t),\omega :=2\pi \mathrm{.}$

Then $\xi$ is a longitudinal plane wave with speed $\sqrt{(\lambda +2\mu )\mathrm{/}{\rho }_0}=2$, and the associated strain and stress satisfy the elastic momentum identity; the strain energy density at time $t$ equals both $W(e)=2{\pi }^2{\sin }^2(\pi x_1-\omega t)$ and the decomposition $\mu \mathrm{\mid }{e}_0{\mathrm{\mid }}_F^2+(\lambda \mathrm{/}2+\mu \mathrm{/}3)(\mathrm{tr}e{)}^2$ with the same value.

• The example is a finite parametric check of the theorem and of the energy identity; it does not show that every elastic solution is such a wave, and no material is identified with these Lamé parameters.

Let $L>0$, $c>0$ and define the odd $2L$-periodic $C^{\mathrm{\infty }}$ datum

$F(x):=\sin \left(\frac{\pi x}{L}\right),G(x):=\frac{c\pi }{L}\sin \left(\frac{\pi x}{L}\right)\mathrm{.}$

Then the sine coefficients of Sine coefficients, sine series and the odd periodic extension are $b_1(F)=1$ and $b_n(F)=0$ for $n\ge 2$, and $b_1(G)=c\pi \mathrm{/}L$, $b_n(G)=0$ for $n\ge 2$; the series solution of the string theorem is the finite expression

$u(t,x)=[\cos ({\omega }_{1}t)+\sin ({\omega }_{1}t)]\sin \left(\frac{\pi x}{L}\right),{\omega }_1=\frac{c\pi }{L},$

which solves $u_{tt}=c^2{u}_{xx}$ with $u(0,\cdot )=F$, $u_t(0,\cdot )=G$ and $u(t,0)=u(t,L)=0$; its tail beyond the first mode vanishes. For each $N\ge 1$ define the matching partial sum $u_N(t,x):={\sum }_{n=1}^N[a_n\cos ({\omega }_{n}t)+d_n\sin ({\omega }_{n}t)]\sin (n\pi x\mathrm{/}L)$, where $a_n=b_n(F)$, $d_n=b_n(G)\mathrm{/}{\omega }_n$ and ${\omega }_n=cn\pi \mathrm{/}L$. The numerical bound of the string theorem at these data is

$\mathrm{\mid }u(t,x)-u_N(t,x)\mathrm{\mid }\le 8(N+1{)}^{-4}(N\ge 1),$

so the general bound is valid but far from sharp on this datum.

• The finite datum exercises the theorem without any infinite tail; the numerical bound $8(N+1{)}^{-4}$ is valid because the general constant is valid, and the example shows explicitly that the general bound is not claimed to be sharp.

False statement. For every continuous odd $2L$-periodic function $F$ there is a constant $C_F$ such that, for every $N\ge 1$,

${\sup }_{x\in \mathbb{R}}\mid F(x)-S_N^{\mathrm{s}}F(x)\mid \le C_F(N+1{)}^{-4},$

where $S_N^{\mathrm{s}}F$ is the $N$-th partial sum of the sine series of Sine coefficients, sine series and the odd periodic extension.

(The theorem of this collection proves the bound only under the $C^5$ hypothesis of The classical $C^2$ fixed-end string: the sine series solves the wave equation with the given data, uniquely, with an explicit tail bound; the statement above drops that hypothesis, and that is exactly what fails.)

• The datum is $C^1$ and is not $C^5$: otherwise the coefficient-decay lemma would bound $n^{-3}$ by a fixed multiple of $n^{-5}$ for every $n$, which is impossible. Nonabsolute convergence of a formally differentiated series alone would not prove failure of $C^2$ regularity. The counterexample refutes the hypothesis-free estimate; it does not claim $C^5$ is necessary for every datum having an $O(N^{-4})$ tail.

• The lower bound uses the $L^2$ tail of the sine coefficients; it shows the tail cannot be better than order $N^{-5\mathrm{/}2}$ in sup norm for this datum, while any $O(N^{-4})$ assertion would require the considerably faster decay of the $C^5$ class.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the cited Lebesgue weak/entropy formulation.

Take $u_L=3>u_R=1$ in the shock of Riemann shock and rarefaction waves for $u_t+(u^2\mathrm{/}2{)}_x=0$ are weak solutions, and the shock is an entropy solution. The shock speed is $s=\frac{u_L+u_R}{2}=2$. For the entropy pair $({\eta }_k,q_k)$:

• for $k=2$, which lies strictly between $u_R=1$ and $u_L=3$, the interface coefficient is $s[{\eta }_2]-[q_2]=1$;
• for $k=0$, which lies outside the interval $[1,3]$, the coefficient is $s[{\eta }_0]-[q_0]=0$;

both nonnegative, in accordance with the theorem's closed formula $s[{\eta }_k]-[q_k]=-(k-u_L)(k-u_R)$ in the interior case and $=0$ in the exterior case.

For the rarefaction with $u_L=1<u_R=3$, the fan $u=x\mathrm{/}t$ satisfies $u_t+u{u}_x=0$ identically, and the entropy residuals vanish because the fan is continuous.

• This is a finite check of the closed formulas; it does not test the general weak formulation beyond the stated Riemann profiles, and the numerical values $3,1,2,0$ are chosen only to exhibit both the interior and the exterior case of the entropy coefficient.

Take $h=1$, $\eta =1$ in the plane channel of Exact plane Poiseuille and Couette flows of the incompressible Newtonian model, with flux and ansatz uniqueness.

1. Poiseuille, $G=12$. The profile is $v(y)=6y(1-y)$; its flux per unit width is $1$ and its maximal speed is $v(1\mathrm{/}2)=3\mathrm{/}2$.

2. Couette, $V=2$. The profile is $v(y)=2y$; the only nonzero entry of $Du$ is ${\mathrm{\partial }}_y{u}_1=2$, so the antisymmetric part of the velocity gradient has magnitude $1$ in the relevant off-diagonal slots and the flow is not irrotational.

• The numbers $h=\eta =1$ and $G=12$, $V=2$ are chosen to give exact arithmetic; they are not measurements and carry no empirical claim about any fluid.

Take $N=4$, $a=1$, $c=1$ in the cyclic chain of The cyclic spring chain, its discrete energy, and the finite-regularity periodic comparison class. The wave number $\kappa =\pi \mathrm{/}2$ gives

${\omega }^2=\frac{4c^2}{a^2}{\sin }^2\left(\frac{\kappa a}{2}\right)=4{\sin }^2\left(\frac{\pi }{4}\right)=2,\omega =\sqrt{2},$

and the profile $q_j(t)=\cos (\frac{\pi j}{2}-\sqrt{2}t)$, $j\in \mathbb{Z}\mathrm{/}4$, solves the chain equation exactly; the continuum comparison at this wave number gives the exact error $\mathrm{\mid }{\omega }^2-c^2{\kappa }^2\mathrm{\mid }=\frac{{\pi }^2}{4}-2$ against the exact bound $\frac{c^2{\kappa }^4{a}^2}{12}=\frac{{\pi }^4}{192}$.