A Global Multidimensional Shock Wave for the Steady Supersonic Isothermal Flow

Abstract

In the reference (Cui and Yin, Pacific J. Math. 233:257–289, 2007), under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, the authors have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic flow past a circular cone. In this paper, we will remove all the symmetric assumptions in the previous paper and study the global existence problem on a really multidimensional shock wave. More concretely, we establish the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic isothermal flow past an infinitely long conic body.

Appendix

Proof of Lemma 3.2

First we estimate P ₅(s) and $P_{1}'(s)$. By (3.3), Lemmas 2.1–2.2 and Remark 2.1,

$$\begin{aligned} P_5(s) =&\frac {1}{{\hat{u}}_z^2(s)-A} \biggl(s \hat{u}_r(s) \hat{u}_r'(s) +s \hat{u}_z(s) \hat{u}_z'(s) -\frac {A}{2} \biggr) \\ =&\biggl\{ \bigl(b_0+O\bigl(e^{-n_0 q_0^{2}}\bigr)\bigr) \frac {b_0q_0}{1+b_0^2}\frac {-q_0}{(1+b_0^2)^2}\bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr)\bigr) \\ &{}-\bigl(b_0^2+O\bigl(e^{-n_0 q_0^{2}}\bigr)\bigr) \frac {q_0}{1+b_0^2}\frac {-q_0}{(1+b_0^2)^2}\bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr)\bigr) - \frac {A}{2}\biggr\} \\ &{}\times\biggl\{ \frac {q_0^2}{(1+b_0^2)^2}\bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr) \bigr)-A\biggr\} ^{-1} \\ =&\frac {-\frac {A}{2}((1+b_0^2)^2)}{q_0^2-A(1+b_0^2)^2}\bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr)\bigr) \\ =&-\frac {A(1+b_0^2)^2}{2q_0^2}+O\bigl(q_0^{-4}\bigr). \end{aligned}$$

Next we derive an expression for $P_{1}'(s)$. Since

$$P_1(s)=\frac {\hat{u}_{z}(s)\hat{u}_{r}(s)}{\hat{u}_z^2(s)-A}, $$

then

$$P_1'(s)= \frac {\hat{u}_z'(s)\hat{u}_r(s)+\hat{u}_z(s)\hat{u}_r'(s)}{\hat{u}_z^2(s)-A} -\frac {2\hat{u}_r(s)\hat{u}_z(s)^2\hat{u}_z'(s)}{(\hat{u}_z^2(s)-A)^2}. $$

It follows from Lemmas 2.1–2.2, Remark 2.1 and a direct computation that

$$\begin{aligned} P_1'(s) =&\frac {\frac {b_0 q_0}{(1+b_0^2)^2}\frac {b_0 q_0}{1+b_0^2} (1+O(e^{-n_0 q_0^{2}}) )-\frac {q_0}{(1+b_0^2)^2} \frac {q_0}{1+b_0^2} (1+O(e^{-n_0 q_0^{2}}) )}{\frac {q_0^2}{(1+b_0^2)^2} (1+O(e^{-n_0 q_0^{2}}) ) -A} \\ &{}-\frac {2(\frac {q_0}{1+b_0^2})^2\frac {b_0q_0}{1+b_0^2}\frac {b_0q_0}{(1+b_0^2)^2} (1+O(e^{-n_0 q_0^{2}}) )}{ (\frac {q_0^2}{(1+b_0^2)^2} (1+O(e^{-n_0 q_0^{2}}) ) -A)^2} \\ =&\frac {(b_0^2-1)(1-\frac {A(1+b_0^2)^2}{q_0^2})-2b_0^2}{(1+b_0^2)(1-\frac {A(1+b_0^2)^2}{q_0^2})^2} \bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr) \bigr) \\ =&-1-\frac {A(1+b_0^2)(1+3b_0^2)}{q_0^2}+O\bigl(q_0^{-4}\bigr). \end{aligned}$$

Other estimates in Lemma 3.2 can be carried out analogously in terms of the expressions for P _i , 1≤i≤5, in (3.3) and Lemmas 2.1 and 2.2; we omit the details here. The proof of Lemma 3.2 is completed. □

Next, we provide the proof of Lemma 3.3.

Proof of Lemma 3.3

From the expressions for B _i , i=1,2,3, in (3.6) and Lemmas 2.1 and 2.2, one has

$$\begin{aligned} B_1 =& \biggl(\frac {\hat{\rho}(s_0)b_0q_0}{A(1+b_0^2)} \biggl(\frac {b_0^2q_0^2}{(1+b_0^2)^2} + \frac {q_0}{(1+b_0^2)} \biggl(\frac {q_0}{(1+b_0^2)}-q_0 \biggr) \biggr) +2 \frac {\hat{\rho}(s_0)b_0q_0}{1+b_0^2} \biggr) \\ &{}\times \bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr) \bigr) \\ =&\frac {2\hat{\rho}(s_0)b_0q_0}{1+b_0^2} \bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr) \bigr). \end{aligned}$$

Similarly,

$$\begin{aligned} &B_2=\frac {(1-b_0^2)\hat{\rho}(s_0)q_0}{1+b_0^2} \bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr) \bigr), \\ &B_3=-\frac {\hat{\rho}(s_0)b_0q_0^2}{(1+b_0^2)^2} \bigl(1+O\bigl(e^{-n_0 q_0^{2}}\bigr) \bigr). \end{aligned}$$

On the other hand, expressions for μ _i (i=1,2) can be obtained from the estimates of B _i which completes the proof of Lemma 3.3. □