Stability of Hydraulic Shock Profiles

Abstract

We establish nonlinear $H^2\cap L^1 \rightarrow H^2$ stability with sharp rates of decay in $L^p$, $p\ge 2$, of general hydraulic shock profiles, with or without subshocks, of the inviscid Saint-Venant equations of shallow water flow, under the assumption of Evans–Lopatinsky stability of the associated eigenvalue problem. We verify this assumption numerically for all profiles, giving in particular the first nonlinear stability results for shock profiles with subshocks of a hyperbolic relaxation system.

Appendices

Appendix A. Decomposition Map

The decomposition of Green kernel function G can be summarized as

$$\begin{aligned}&G=\chi _{|x-y|/t<S}\left( I+II+III\right) ,\quad I=\chi _{t\le 1}I+\chi _{t> 1}I,\nonumber \\&\chi _{t> 1}I=\chi _{t> 1}(I^1+I^2),\nonumber \\&I^1=I^1_S+I^1_R=I_{S1}^1+I_{S2}^1+I_{S3}^1+I_{R1}^1+I_{R2}^1+I_{R3}^1,\nonumber \\&I^2=I^2_{R1}+I_{R2}^2+I_{R3}^2,\nonumber \\&I_{S2}^1=\chi _{\frac{\bar{\alpha }}{p}>\varepsilon }I_{S2}^1+\chi _{\frac{\bar{\alpha }}{p}\le \varepsilon }\left( S^1+I_{S2Ri}^1+I_{S2Rii}^1\right) ,\\&I_{R2}^{1,2}=\chi _{\frac{\bar{\alpha }}{p}>\varepsilon }I_{R2}^{1,2}+\chi _{\frac{\bar{\alpha }}{p}\le \varepsilon }I_{R2}^{1,2},\nonumber \\&III=III^1+III^2=III^1_a+III^1_b+III^1_c+III^2_a+III^2_b+III^2_c,\nonumber \\&III^{1,2}_a:=H^{1,2},\nonumber \end{aligned}$$

(A.1)

in which we see

$$\begin{aligned} \begin{aligned}&H^{1,2}=\chi _{|x-y|/t<S}III^{1,2}_a,\quad S^1=\chi _{|x-y|/t<S,t>1,\bar{\alpha }/p\le \varepsilon }S^1,\\&R=\chi _{|x-y|/t<S}\left( II+III^{1,2}_{b,c}+\chi _{t\le 1}I\right. \\&\quad \left. +\chi _{t> 1}\left( I^1_{S1,S3,R}+\chi _{\frac{\bar{\alpha }}{p}>\varepsilon }I_{S2}^1+\chi _{\frac{\bar{\alpha }}{p}\le \varepsilon }\left( I_{S2Ri}^{1}+I_{S2Rii}^1\right) +I^2\right) \right) . \end{aligned} \end{aligned}$$

(A.2)

Appendix B. Integral Estimates

$I^1_{S2Ri}:$ Setting $f(u)=\frac{1}{\sqrt{4c^2_{2,-}\pi u}}e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-} u}}$, yields $I^1_{S2Ri}=f(\frac{t}{c^1_{2,-}})-f(x-y)$, in which by (7.32) $\frac{t}{c^1_{2,-}}$ and $x-y$ are comparable. Writing the difference as an integral yields

$$\begin{aligned} |I^1_{S2Ri}|=\frac{1}{\sqrt{4c^2_{2,-}\pi }}\left| \int _{x-y}^{\frac{t}{c^1_{2,-}}}e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}u}}\frac{\frac{(t-c^1_{2,-}(x-y))^2}{2c^2_{2,-}}-u}{2u^{\frac{5}{2}}}du\right| . \end{aligned}$$

(B.1)

Using that $\frac{t}{c^1_{2,-}}$ and $x-y$ are comparable, we have $e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}u}}\le e^{-\frac{c^1_{2,-}(t-c^1_{2,-}(x-y))^2}{8 c^2_{2,-}t}}$, which, together with $x^ne^{-x^2}\lesssim e^{-\frac{x^2}{2}}$ for any n positive, yields

$$\begin{aligned}&|I^1_{S2Ri}|\lesssim \int _{x-y}^{\frac{t}{c^1_{2,-}}}\left| e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}u}}\frac{\frac{(t-c^1_{2,-}(x-y))^2}{2c^2_{2,-}}-u}{2u^{\frac{5}{2}}}\right| |du|\nonumber \\&\quad \lesssim e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}\frac{2t}{c^1_{2,-}}}}(t-c^1_{2,-}(x-y))^2\left| \int _{x-y}^{\frac{t}{c^1_{2,-}}}u^{-\frac{5}{2}}du\right| +e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}\frac{2t}{c^1_{2,-}}}}\left| \int _{x-y}^{\frac{t}{c^1_{2,-}}}u^{-\frac{3}{2}}du\right| \nonumber \\&\quad \lesssim e^{-\frac{c^1_{2,-}(t-c^1_{2,-}(x-y))^2}{8c^2_{2,-}t}}(t-c^1_{2,-}(x-y))^2\left| \frac{1}{\left( \frac{t}{c^1_{2,-}}\right) ^{1.5}}-\frac{1}{\left( x-y\right) ^{1.5}}\right| \nonumber \\&\quad +e^{-\frac{c^1_{2,-}(t-c^1_{2,-}(x-y))^2}{8c^2_{2,-}t}}\left| \frac{1}{\left( \frac{t}{c^1_{2,-}}\right) ^{0.5}}-\frac{1}{\left( x-y\right) ^{0.5}}\right| \nonumber \\&\quad \lesssim e^{-\frac{c^1_{2,-}(t-c^1_{2,-}(x-y))^2}{8c^2_{2,-}t}}\frac{(t-c^1_{2,-}(x-y))^3}{t^{2.5}}+e^{-\frac{(t-c^1_{2,-}(x-y))^2}{8c^2_{2,-}t}}\frac{(t-c^1_{2,-}(x-y))}{t^{1.5}}\nonumber \\&\quad \lesssim e^{-\frac{c^1_{2,-}(t-c^1_{2,-}(x-y))^2}{16c^2_{2,-}t}}\frac{1}{t}. \end{aligned}$$

(B.2)

$\frac{\partial I^1_{S2Ri}}{\partial y}:$

$$\begin{aligned}&|\frac{\partial I^1_{S2Ri}}{\partial y}|=\frac{1}{\sqrt{4c^2_{2,-}\pi }}\left| \frac{\partial }{\partial y}\int _{x-y}^{\frac{t}{c^1_{2,-}}}e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}u}}\frac{\frac{(t-c^1_{2,-}(x-y))^2}{2c^2_{2,-}}-u}{2u^{\frac{5}{2}}}du\right| \nonumber \\&\quad \lesssim \left| e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}(x-y)}}\frac{\frac{(t-c^1_{2,-}(x-y))^2}{2c^2_{2,-}}-(x-y)}{2(x-y)^{\frac{5}{2}}}\right| \nonumber \\&\quad +e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}\frac{2t}{c^1_{2,-}}}}(t-c^1_{2,-}(x-y))^3\left| \int _{x-y}^{\frac{t}{c^1_{2,-}}}u^{-\frac{7}{2}}du\right| \nonumber \\&\quad +e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}\frac{2t}{c^1_{2,-}}}}(t-c^1_{2,-}(x-y))\left| \int _{x-y}^{\frac{t}{c^1_{2,-}}}u^{-\frac{5}{2}}du\right| \nonumber \\&\quad \lesssim e^{-\frac{c^1_{2,-}(t-c^1_{2,-}(x-y))^2}{16c^2_{2,-}t}}\frac{1}{t^{1.5}}. \end{aligned}$$

(B.3)

$I^1_{S2Rii}:$ Using that $e^{-\frac{(t-c^1_{2,-}(x-y))^2}{4c^2_{2,-}(x-y)}}<1$ and that for the complementary error function $erfc(x):=\frac{2}{\sqrt{\pi }}\int _x^{\infty }e^{-z^2}\mathrm{d}z$, there is the estimate $erfc(x)\le {e^{-x^2}}$, $I^1_{S2Rii}$ can be bounded by

$$\begin{aligned}&|I^1_{S2Rii}|\lesssim \int _r^\infty e^{-c^2_{2,-}(x-y)\xi ^2}\mathrm{d}\xi =\frac{1}{\sqrt{(x-y)c^2_{2,-}}}erfc(\sqrt{c^2_{2,-}(x-y)}r)\nonumber \\&\quad \lesssim e^{-r^2c^2_{2,-}(x-y)}\le e^{-r^2c^2_{2,-}\frac{t}{2}}, \end{aligned}$$

(B.4)

in which we have used that $x-y$ is comparable to t hence is bounded away from 0 and is greater than $\frac{t}{2}$. Term $I_{S2Rii}$ is then time-exponentially small.

$\frac{\partial I^1_{S2Rii}}{\partial y}:$ When the partial derivative hits the exponential outside the integral we get time- exponentially small terms by following the proof for $I^1_{S2Rii}:$. When the partial derivative hits inside the integral we use $x^2e^{-x^2}\lesssim e^{-x^2/2}$ and again get time-exponentially small terms by following the proof for $I^1_{S2Rii}.$

$I^1_{R2i}:$ Using that $xe^{-x^2}\lesssim e^{\frac{-x^2}{2}}$ and that $\frac{t}{c^1_{2,-}}$ and $x-y$ are comparable ($\frac{t}{2c^1_{2,-}}<x-y<\frac{2t}{c^1_{2,-}}$), we have

$$\begin{aligned}&e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}O|\eta _*|\mathrm{d}\xi \nonumber \\&\quad \lesssim e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}\frac{|t-c^1_{2,-}(x-y)|}{x-y}\mathrm{d}\xi \nonumber \\&\quad \lesssim \frac{|t-c^1_{2,-}(x-y)|}{(x-y)^{\frac{3}{2}}}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\lesssim \frac{1}{(x-y)}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\left( x-y\right) }}\nonumber \\&\quad \le \frac{1}{\frac{t}{2c^1_{2,-}}}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\frac{2t}{c^1_{2,-}}}}, \end{aligned}$$

(B.5)

$$\begin{aligned}&e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}O|\xi |\mathrm{d}\xi \nonumber \\&\quad \lesssim e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{0}^r e^{-\xi ^2c^2_{2,-}(x-y)}\xi \mathrm{d}\xi \nonumber \\&\quad \lesssim \frac{1}{x-y}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}(1-e^{-r^2c^2_{2,-}(x-y)})\le \frac{1}{\frac{t}{2c^1_{2,-}}}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\frac{2t}{c^1_{2,-}}}}, \end{aligned}$$

(B.6)

$$\begin{aligned}&e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}O|\eta _*^3(x-y)|\mathrm{d}\xi \nonumber \\&\quad \lesssim e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}\frac{|t-c^1_{2,-}(x-y)|^3}{(x-y)^2}\mathrm{d}\xi \nonumber \\&\quad \lesssim \frac{|t-c^1_{2,-}(x-y)|^3}{(x-y)^{\frac{5}{2}}}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\nonumber \\&\quad \lesssim \frac{1}{(x-y)}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\left( x-y\right) }}\lesssim \frac{1}{t}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\frac{2t}{c^1_{2,-}}}}, \end{aligned}$$

(B.7)

$$\begin{aligned}&e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}O|\eta _*^2\xi (x-y)|\mathrm{d}\xi \nonumber \\&\quad \lesssim e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\frac{|t-c^1_{2,-}(x-y)|^2}{x-y}\int _{0}^r e^{-\xi ^2c^2_{2,-}(x-y)}\xi \mathrm{d}\xi \nonumber \\&\quad \lesssim \frac{|t-c^1_{2,-}(x-y)|^2}{(x-y)^2}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}(1-e^{-r^2c^2_{2,-}(x-y)})\nonumber \\&\quad \lesssim \frac{1}{t}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\frac{2t}{c^1_{2,-}}}}, \end{aligned}$$

(B.8)

$$\begin{aligned}&e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}O|\eta _*\xi ^2(x-y)|\mathrm{d}\xi \nonumber \\&\quad \lesssim |t-c^1_{2,-}(x-y)|e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _0^r e^{-\xi ^2c^2_{2,-}(x-y)}\xi ^2\mathrm{d}\xi \nonumber \\&\quad \lesssim \frac{|t-c^1_{2,-}(x-y)|}{(x-y)^{\frac{3}{2}}}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\nonumber \\&\quad \lesssim \frac{1}{(x-y)}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\left( x-y\right) }}\nonumber \\&\quad \lesssim \frac{1}{t}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\frac{2t}{c^1_{2,-}}}}, \end{aligned}$$

(B.9)

$$\begin{aligned}&e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{-r}^r e^{-\xi ^2c^2_{2,-}(x-y)}O|\xi ^3(x-y)|\mathrm{d}\xi \nonumber \\&\quad \lesssim (x-y)e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\int _{0}^r e^{-\xi ^2c^2_{2,-}(x-y)}\xi ^3 \mathrm{d}\xi \nonumber \\&\quad \lesssim \frac{1}{x-y}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{4c^2_{2,-}\left( x-y\right) }}\nonumber \\&\quad \lesssim \frac{1}{t}e^{-\frac{\left( t-c^1_{2,-}(x-y)\right) ^2}{8c^2_{2,-}\frac{2t}{c^1_{2,-}}}}. \end{aligned}$$

(B.10)

We then see that all terms are absorbable in R (7.6).

Appendix C. Computational Framework

1.1 C.1. Computational Environment

In carrying out our numerical investigations, we have used MacBook Pro 2017 with 16GB memory and Intel Core i7 processor with 2.8GHz processing speed for coding and debugging. The main parallelized computation is done in the compute nodes of IU Karst, a high-throughput computing cluster. It has 228 compute nodes. Each node is an IBM NeXtScale nx360 M4 server equipped with two Intel Xeon E5-2650 v2 8-core processors and with 32 GB of RAM and 250 GB of local disk storage.

1.2 C.2. Computational Time

The computational times displayed in the tables below are times elapsed in a single processor of IU Karst.

Table 1 Times to compute a single Evans–Lopatinsky determinant $\Delta _{F,H_R}(\lambda )$

Table 2 Times to compute a single Evans determinant $D_{F,H_R}(\lambda )$