12. Statement of supersonic transport BVP: uniqueness of solutions

We suppose here that supersonic transport loads, moving at supersonic velocity c>c1, are known on the boundary D:

$$P = \sigma\_{\vec{\eta}} n\_i e\_{\vec{\eta}} = \rho c^2 p\_{\vec{\eta}}(\mathbf{x}, z) e\_{\vec{\eta}} \theta(z), \mathbf{x} = (\mathbf{x}\_1, \mathbf{x}\_2) \in \mathbb{S}, \qquad \mathbf{i}, \mathbf{j} = 1, 2, 3 \tag{56}$$

Functions pj(x,z) are integrable on D+. We assume here G = 0 and

$$
\mu(\mathbf{x}, z) = 0, \quad \mu\_{i \cdot z}(\mathbf{x}, z) = 0, \quad z \le 0, \quad \mathbf{x} \in \mathbb{S}^- \tag{57}
$$

For ∥(x,z)∥ ! ∞

$$\mu\_j \to 0, \exists \varepsilon > 0:\quad \|\partial\_j u\| < O(\|(x, z)\|^{1+\varepsilon}), \qquad j = 1, 2, z \tag{58}$$

The jump conditions, Eqs. (15) and (16) are satisfied on the shock wave fronts.

Theorem 12.1. The solution of the supersonic transport boundary value problem is unique.

Proof. Suppose that there exist two solutions. Since the problem is linear, it follows that their difference u(x,z) satisfies the zero boundary conditions, i.e., P(x;z) = 0, and is a solution of the homogeneous equations of motion (G = 0). We note, that Lemma 8.1 is also true in the supersonic case for shock waves as there is Theorem 3.2 for the gaps of energy on their fronts (see full proof). Then together with conditions given in Eq. (59) of decay of the solutions at infinity and the zero conditions for z = 0,

$$\int\_{S^{-}} E(\mathbf{x}, z) d\mathbf{x}\_1 d\mathbf{x}\_2 = \int\_{S^{-}} \sigma\_{l3} u\_{l^\*z}(\mathbf{x}, z) d\mathbf{x}\_1 d\mathbf{x}\_2 \to 0 \quad by \quad z \to \infty$$

The energy density E is a positive definite quadratic form of ui,j by construction. Therefore, by virtue of the decay of the solution at infinity, the relation only holds if ui,j = 0 for all i and j. Hence, we obtain u = 0; i.e., the solutions coincide. The proof of the theorem is complete.

Theorem 12.1 holds for both exterior and interior boundary value problems.

Since the tensors His

146 Differential Equations - Theory and Current Research

sense.

say that they are stationary. Accordingly, the tensors Hid

δ j i H�

þ ð

sonic transport boundary value problem.

known on the boundary D:

For ∥(x,z)∥ ! ∞

S

P ¼ σijniej ¼ rc

2 pj

Functions pj(x,z) are integrable on D+. We assume here G = 0 and

The jump conditions, Eqs. (15) and (16) are satisfied on the shock wave fronts.

Theorem 12.1. The solution of the supersonic transport boundary value problem is unique.

Proof. Suppose that there exist two solutions. Since the problem is linear, it follows that their difference u(x,z) satisfies the zero boundary conditions, i.e., P(x;z) = 0, and is a solution of the homogeneous equations of motion (G = 0). We note, that Lemma 8.1 is also true in the supersonic case for shock waves as there is Theorem 3.2 for the gaps of energy on their fronts

For this type of tensors, the next theorem was proved (see [7]).

Theorem 11.1. The fundamental stress tensor H satisfies the relation

<sup>S</sup> ð Þx θð Þ¼ z

rc <sup>2</sup> � ��<sup>1</sup> Σ~j

jkð Þx independent of z inside the sonic cones Kl (l = 1,2), we conventionally

ð Þ y � x; τ; n yð Þ dS yð Þþ

dy1dy<sup>2</sup>

ð Þ x; z ejθð Þz , x ¼ ð Þ x1; x<sup>2</sup> ∈S, i, j ¼ 1, 2, 3 (56)

u xð Þ¼ ; z 0, ui, <sup>z</sup>ð Þ¼ x; z 0, z ≤ 0, x∈ S� (57)

uj ! <sup>0</sup>, <sup>∃</sup><sup>ε</sup> <sup>&</sup>gt; <sup>0</sup> : <sup>∥</sup>∂ju<sup>∥</sup> <sup>&</sup>lt; <sup>O</sup> <sup>∥</sup>ð Þ <sup>x</sup>; <sup>z</sup> <sup>∥</sup><sup>1</sup>þ<sup>ε</sup> � �, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, z (58)

, <sup>z</sup>ðy � x; zÞ

they depend essentially on z, although they are regular functions. The aforementioned sym-

Hj i

For x∉D all integrals are regular; for x ∈ D the first integral is singular, calculated in value principle

This theorem enables us to obtain solvable singular boundary integral equations for a super-

We suppose here that supersonic transport loads, moving at supersonic velocity c>c1, are

i

metry properties hold for both stationary and dynamic terms in the tensors.

ðz

0 dτ ð

S

12. Statement of supersonic transport BVP: uniqueness of solutions

<sup>i</sup>3ð<sup>x</sup> � <sup>y</sup>; <sup>z</sup>Þ � <sup>U</sup><sup>j</sup>

� �

<sup>j</sup> ð Þ x; z are said to be dynamic, because

#### 13. Statement of supersonic BVP in D<sup>0</sup> <sup>3</sup> <sup>R</sup><sup>3</sup> � � and its generalized solution

To solve the problem, we also use the method of generalized functions. We introduce here the regular generalized function with support on D� þ:

$$
\widehat{\boldsymbol{\mu}}\_{\circ}(\mathbf{x}, \boldsymbol{z}) = \boldsymbol{\mu}\_{\circ}(\mathbf{x}, \boldsymbol{z}) H\_{\mathcal{S}}^{-}(\mathbf{x}) \boldsymbol{\theta}(\mathbf{z}) \tag{59}
$$

Also using the properties of differentiation of regular generalized functions with gaps at D, and taking into account the boundary conditions and the conditions on the fronts, we obtain the transport Lame equations (Eq. (14)) on the space of distributions with singular mass forces:

$$
\hat{\mathbf{g}}\_{\rangle} = p\_{\rangle} \delta\_{\mathcal{S}}(\mathbf{x}) \theta(\mathbf{z}) + \left( \left( \lambda \mu\_k n\_k \delta\_{\vec{\eta}} + \mu \left( \mu\_i n\_{\vec{\eta}} + \mu\_j n\_i \right) \right) \delta\_{\mathcal{S}}(\mathbf{x}) \theta(\mathbf{z}) \right),
\tag{60}
$$

By using the properties of convolutions with the Green tensor and the boundary conditions, we obtain the generalized solution of BVP in the form:

$$\rho c^2 \widehat{\boldsymbol{u}}\_k = \widehat{\boldsymbol{L}}\_k^j \ast \boldsymbol{P}\_{\overline{\boldsymbol{\beta}}} \boldsymbol{\delta} \mathbf{s}(\mathbf{x}) \boldsymbol{\theta}(\mathbf{z}) + \widehat{\boldsymbol{L}}\_{k'}^j \boldsymbol{i}^\* \left(\lambda \boldsymbol{u}\_m \boldsymbol{n}\_m \delta\_{\overline{\boldsymbol{\beta}}} + \mu \left(\boldsymbol{u}\_i \boldsymbol{n}\_{\overline{\boldsymbol{\beta}}} + \boldsymbol{u}\_j \boldsymbol{n}\_i\right)\right) \boldsymbol{\delta} \mathbf{s}(\mathbf{x}) \boldsymbol{\theta}(\mathbf{z}) \tag{61}$$

By analog with the subsonic case, if we use fundamental stress tensor, then the right-hand side of Eq. (61) may be represented in the form of a surface integral over the boundary of the domain. In our notation, on the boundary, it acquires the form

$$\mu\_i H\_S^-(\mathbf{x})\theta(\mathbf{z}) = \int\_{D\_+} \left( \mathcal{U}\_i^j(\mathbf{x}, y, z - \tau) \, p\_j(y, \tau) - T\_i^j(\mathbf{x}, y, z - \tau, n(y, \tau)) u\_j(y, \tau) \right) dD(y, \tau) \tag{62}$$

This formula is similar to the Somigliana formula in the static theory of elasticity ([1]: 146), but it is impossible to use this formula to determine the solution of the boundary value problem in the case of supersonic loads, because the second term contains strong non-integrable singularities of the tensor T on the shock wave fronts of fundamental solutions; therefore, the integrals are divergent. To construct a regular integral representation of the formula, we must regularize it. For this, we use the tensor H.
