Theorem 1. 1. a) Let conditions (2.2) hold and $\frac{1}{2}\operatorname{div}\mathbf B-C\leqslant c_0I_m$ almost everywhere (a.e.) in $\Pi_T$ for a constant $c_0>0$ (we can assume that $c_0\leqslant\|\frac{1}{2}\operatorname{div}\mathbf B-C\|_{L^\infty(\Pi_T)}$). Then there exists a weak solution $\mathbf w$ of the Cauchy problem (2.1) such that

b) In addition, if $\nabla\mathbf B\in L^\infty(\Pi_T)$ and $\displaystyle \lim_{\xi\to 0}\int_0^T\|\Delta_\xi C\|_{L^\infty(\mathbb{R}^n)}\,dt=0$, then the weak solution is unique. It also has the property $\mathbf w\in C(0,T;L^2(\mathbb{R}^n))$; therefore, the above estimate takes the form

2. Let $\|\{\mathbf B,C,\nabla\mathbf B,\nabla C\}\|_{L^\infty(\Pi_T)}\leqslant N$, $\mathbf f,\nabla\mathbf f\in L^{2,1}(\Pi_T)$ and $\mathbf w_0\in H^1(\mathbb{R}^n)$. Then the weak solution $\mathbf w$ is a strong solution, which is unique and satisfies the estimate

If also $\mathbf f\in L^{2,q}(\Pi_T)$ for some $1\leqslant q\leqslant\infty$, then $\partial_t\mathbf w\in L^{2,q}(\Pi_T)$ and

3. a) Let, in addition, $\|\{\nabla^2\mathbf B,\nabla^2C\}\|_{L^\infty(\Pi_T)}\leqslant N$, $\nabla^2\mathbf f\in L^{2,1}(\Pi_T)$ and $\mathbf w_0\in H^2(\mathbb{R}^n)$. Then the strong solution $\mathbf w$ has $\nabla^2\mathbf w \in L^{2,\infty}(\Pi_T)$, $\partial_t\nabla\mathbf w \in L^{2,1}(\Pi_T)$, and

b) If also $\|\{\partial_t\mathbf B,\partial_tC\}\|_{L^\infty(\Pi_T)}\leqslant N$ and $\mathbf f,\nabla\mathbf f,\partial_t\mathbf f\in L^{2,q}(\Pi_T)$ for some $q$, ${1\leqslant q\leqslant\infty}$, then, in addition, there exists $\partial_t^2\mathbf w\in L^{2,q}(\Pi_T)$ and

First of all, we recall that the property $B_i=B_i^T$, $i=1,\dots,n$, and integration by parts with respect to $x$ (that is, with respect to $x_1,\dots,x_n$) yield $(B_i\partial_i\mathbf v,\mathbf v)_{\mathbb{R}^n}=-(\mathbf v,(\operatorname{div}\mathbf B)\mathbf v+B_i\partial_i\mathbf v)_{\mathbb{R}^n}$, which implies that

To prove the existence of the weak solution, it is convenient to use the artificial viscosity method (see, for example, [4], Section 7.3) by introducing the following Cauchy problem for a strongly parabolic system of equations with the simplest principal part:

Due to this estimate there exists a function $\mathbf w\in L^{2,\infty}(\Pi_T)$ such that ${\mathbf y_{\tau_k}\to\mathbf w}$ $*$-weakly in $L^{2,\infty}(\Pi_T)$ and $\tau_k\nabla\mathbf y_{\tau_k}\to 0$ in $L^2(\Pi_T)$ for some sequence $\tau_k\to 0$. For $\mathbf w$, estimate (2.4) holds. Taking the limit in the integral identity (2.13) for $\mathbf y=\mathbf y_{\tau_k}$ (using that $(B_i\partial_i\mathbf y,\boldsymbol\varphi)_{\Pi_T}=-(\mathbf y,B_i\partial_i\boldsymbol\varphi+(\operatorname{div}\mathbf B)\boldsymbol\varphi)_{\Pi_T}$) leads to the integral identity (2.3), that is, the function $\mathbf w$ constructed is a weak solution of the Cauchy problem (2.1).

b) We establish the uniqueness of a weak solution under the assumptions in part 1, b). With this aim in view, we obtain estimate (2.4) for any weak solution $\mathbf w$. It suffices to assume that $\mathbf f=0$ and $\mathbf w_0=0$. We use the Friedrichs method (see [12], Ch. 6, § 2). We substitute the function $\boldsymbol\varphi=\sigma^{(h)}\boldsymbol\psi$ into the integral identity (2.3). Transferring $\sigma^{(h)}$ to the other factors (which is possible for averagings with kernel $e_h(\xi)$ that is even with respect to $\xi_1,\dots,\xi_n$) and integrating by parts with respect to $x$, we obtain

An estimate similar to (2.4) holds for $\sigma^{(h)}\mathbf w$. It can be derived like (2.14), but for $\tau=0$; therefore, it is slightly simpler. More precisely, taking the inner product in $L^2(\mathbb{R}^n)$ of the equation in (2.15) and $\sigma^{(h)}\mathbf w$, in view of (2.11) we obtain

We bound the functions $\mathbf f_1^{(h)}$ and $ \mathbf f_2^{(h)}$. Supposing provisionally that $\partial_i\mathbf w\in L_{\mathrm{loc}}^1(\mathbb{R}^n)$, $i=1,\dots,n$, we obtain

c) We introduce the following difference quotient and shift operator with respect to the coordinate $x_j$:

We apply (2.14) to the weak solution $\delta_{jh}\mathbf y_\tau$ of this problem and then use (2.19) under the assumptions on $\mathbf B$ and $C$ from part 2. Therefore, we obtain

It is possible to replace $T$ in this inequality by any $t$, $0<t\leqslant T$; therefore, it is also true that

We have

Integrating by parts with respect to $x$ we write the integral identity (2.3) for $\mathbf w$ in the form

If $\mathbf f\in L^{2,q}(\Pi_T)$ for some $1\leqslant q\leqslant\infty$, then by the equation in (2.1) we have

The assumption on $C$ in part 1, b) is much weaker than $\nabla C\in L^\infty(\Pi_T)$; therefore, the strong solution is unique.

d) Now, under the assumptions in part 1, b) we deduce the property $\mathbf w\in C(0,T; L^2(\mathbb{R}^n))$ of the weak solution, which will complete the proof of part 1. Let $C_k=\sigma^{(1/k)}C$, $k=1,2,\dots$; then $\nabla C_k\in L^\infty(\Pi_T)$ and

Let $\mathbf w_k$ be the strong solution of the Cauchy problem corresponding to $C=C_k$, $\mathbf f=\mathbf f_k$ and $\mathbf w_0=\mathbf w_{0k}$. Then it satisfies an estimate of type (2.4) or, more precisely, of type (2.5):

e) Assume that the assumptions in part 3, a) of the theorem hold. For the strong solution $\mathbf w$, we introduce the functions $\mathbf v_k=\partial_k\mathbf w\in L^{2,\infty}(\Pi_T)$, $k=1,\dots,n$, and obtain an extended system of equations for them (see [13], § 12). We choose $\boldsymbol\varphi:=-\partial_k\boldsymbol\psi$ in the integral identity (2.3), use the formula

This Cauchy problem is similar to the original one. Under the assumptions in part 3, a), we can apply parts 1 and 2 to it. Consequently, its weak solution $\nabla\mathbf w$ is unique, is a strong solution, and satisfies the estimate

If also $\partial_t\mathbf B,\partial_tC\in L^\infty(\Pi_T)$ and $\mathbf f,\nabla\mathbf f,\partial_t\mathbf f\in L^{2,q}(\Pi_T)$ for some $1\leqslant q\leqslant\infty$, then we can apply $\partial_t$ to the equation in (2.1), then use it again and derive the formula

Theorem 2. Assume that conditions (3.2) and (3.4) hold, $\operatorname{div}\mathbf B\in L^\infty(\Pi_T)$, $\frac{1}{2}\operatorname{div}\mathbf B-C\leqslant c_0I_m$ almost everywhere in $\Pi_T$ and $\mathbf g_\tau\in L^2(\Pi_T)$. Then there exists a unique weak solution $\mathbf y=\mathbf y_\tau$ of the Cauchy problem (3.1); it belongs to $C(0,T;L^2(\mathbb{R}^n))$ and satisfies the energy estimate

2. To establish the existence of a solution, we introduce the related initial-boundary value problem

The existence of a weak solution $\mathbf y^{(k)}$ is proved in a standard way using the Galerkin method (see [23], Ch. III). First for the Galerkin approximations and then for $\mathbf y^{(k)}$ itself, after taking the limit, we prove quite similarly to (3.6) that

We set additionally $\mathbf y^{(k)}=0$ on $\Pi_T\setminus\Pi_{kT}$. Then $\mathbf y^{(k)}\in V_2(\Pi_T)$ and inequality (3.8) yields a $k$-uniform estimate similar to (3.6) with $\mathbf y^{(k)}$ in place of $\mathbf y_\tau$ and $L^{2,\infty}(\Pi_T)$ in place of $C(0,T;L^2(\mathbb{R}^n))$. By virtue of this estimate, there exists a function $\mathbf y\in V_2(\Pi_T)$ such that $\mathbf y^{(k)}\to\mathbf y$ $*$-weakly in $L^{2,\infty}(\Pi_T)$ and $\nabla\mathbf y^{(k)}\to \nabla\mathbf y$ weakly in $L^2(\Pi_T)$ for some subsequence $k=k_l\to\infty$. Estimate (3.6) holds for $\mathbf y=\mathbf y_\tau$ with the norm in $L^{2,\infty}(\Pi_T)$ instead of $C(0,T;L^2(\mathbb{R}^n))$.

We rewrite the integral identity (3.7) as (3.5) with $\mathbf y^{(k)}$ playing the role of $\mathbf y$ as follows:

3. We establish that $\mathbf y\in C(0,T;L^2(\mathbb{R}^n))$ and also the uniqueness of the weak solution. To prove the first result we assume provisionally that $\mathbf f_\tau\in L^2(\Pi_T)$. Then for any weak solution $\mathbf y$ we can rewrite (3.5) in the form

For the homogeneous Cauchy problem (that is, the problem with $\mathbf f_\tau=0$, $\mathbf g_\tau=0$, and $\mathbf y_{0\tau}=0$) we choose $\boldsymbol\varphi=\mathbf y$ in $\Pi_t$ and $\boldsymbol\varphi=0$ in $\Pi_T\setminus\Pi_t$, $0<t\leqslant T$ in the last identity. By the formula

It remains to show that $\mathbf y\in C(0,T;L^2(\mathbb{R}^n))$ for $\mathbf f_\tau\in L^{2,1}(\Pi_T)$. With this aim in view (similarly to part d) of the proof of Theorem 1) it suffices to take functions $\mathbf f_{\tau k}\in L^2(\Pi_T)$, $k=1,2,\dots$, such that $\mathbf f_{\tau k}\to\mathbf f_\tau$ in $L^{2,1}(\Pi_T)$ as $k\to\infty$ instead of $\mathbf f_\tau$ and consider the corresponding weak solutions $\mathbf y_k$ of the Cauchy problem. By what we proved above, we have

Theorem 2 is proved.

Theorem 3. Assume that conditions (3.2), (3.4), (3.13), and (3.15) hold and that $\operatorname{div}\mathbf B$, $\partial_tA_{ij}\in L^\infty(\Pi_T)$, $A_{ij}=A_{ji}^T$, $i,j=1,\dots,n$, $\frac{1}{2}\operatorname{div}\mathbf B-C\leqslant c_0I_m$ almost everywhere in $\Pi_T$, $\mathbf y_{0\tau}\in H^1(\mathbb{R}^n)$, $\mathbf y_{1\tau}\in L^2(\mathbb{R}^n)$, and $8\overline{\tau}^2\mu\leqslant 1$. Then there exists a unique weak solution $\mathbf y=\mathbf y_\tau$ of the Cauchy problem (3.10), (3.11), and the following energy estimate holds:

According to the dominance condition (3.13), we have

2. Like in the previous proof, to verify the existence of a solution we introduce the auxiliary initial-boundary value problem

The existence of the weak solution $\mathbf y^{(k)}$ is deduced in a standard way using the Galerkin method (see [24], Ch. IV, Theorem 3.2; the $*$-weak convergence of the Galerkin approximations and their first derivatives in $L^{2,\infty}(\Pi_{kT})$ must be used instead of their weak convergence in $L^2(\Pi_{kT})$). First for the Galerkin approximations and then for $\mathbf y^{(k)}$ itself, after taking the limit, we prove quite similarly to (3.16) that

We set additionally $\mathbf y^{(k)}=0$ on $\Pi_T\setminus\Pi_{kT}$. Then $\mathbf y^{(k)}\in W_{2,\infty}^{1,1}(\Pi_{kT})$, and we can replace $\Omega_k$ by $\mathbb{R}^n$ and $\Pi_{kT}$ by $\Pi_T$ on the left-hand side of (3.20). By this estimate there is a function $\mathbf y\in W_{2,\infty}^{1,1}(\Pi_T)$ such that $\mathbf y^{(k)}\to\mathbf y$, $\nabla\mathbf y^{(k)}\to \nabla\mathbf y$, and $\partial_t\mathbf y^{(k)}\to\partial_t\mathbf y$ $*$-weakly in $L^{2,\infty}(\Pi_T)$ for some subsequence $k=k_l\to\infty$. Thus, estimate (3.16) is true for $\mathbf y=\mathbf y_\tau$.

We rewrite the integral identity (3.19) in the form of (3.12) with $\mathbf y^{(k)}$ in the role of $\mathbf y$:

3. The uniqueness of a weak solution is deduced using the known method (see [24], Ch. IV, Theorem 3.1) that consists in taking $\displaystyle\boldsymbol\varphi(x,t)=\int_t^\theta\mathbf y(x,t')\,dx\,dt'$ in $\Pi_\theta$, $\boldsymbol\varphi(x,t)=0$ in $\Pi_T\setminus\Pi_\theta$, in the integral identity (3.12), where $0<\theta\leqslant T$ is a parameter. This makes it possible to prove that $\mathbf y=0$ for $\mathbf y_{0\tau}=\mathbf y_{1\tau}=0$ and $\mathbf f_\tau=0$. As the arguments are standard, we omit the details. Theorem 3 is proved.

Theorem 4. 1. Assume that the assumptions of Theorem 1, parts 1 and 2 (for $q= 1$), and Theorem 2 hold and that $\|A\|_{L^\infty(\Pi_T)}\leqslant N$. Then the difference $\mathbf r_\tau=\mathbf w-\mathbf y_\tau$ of solutions of the Cauchy problems (2.1) and (3.1) satisfies the estimate

In particular, for $\mathbf f_\tau=\mathbf f$, $\mathbf g_\tau=0$ and $\mathbf y_{0\tau}=\mathbf w_0$ the following simplified estimate is true:

2. Let the assumptions of Theorem 1, part 3, a), also hold, let $\|\operatorname{div} A_{\cdot j}\|_{L^\infty(\Pi_T)}\leqslant N$, $j=1,\dots,n$, where $\operatorname{div} A_{\cdot j}:=\partial_iA_{ij}$ and, again, let $\mathbf f_\tau=\mathbf f$, $\mathbf g_\tau=0$ and $\mathbf y_{0\tau}=\mathbf w_0$. Then $\mathbf r_\tau$ also satisfies the estimate

2. Under the assumptions of part 2 of the theorem the last term on the right-hand side of (4.4) can be replaced by $2\tau\|\partial_i(A_{ij}\partial_j\mathbf w)\|_{L^{2,1}(\Pi_T)}$ (where the sum over $i,j=1,\dots,n$ is assumed under the norm sign). Since, using estimates (2.6) and (2.8) for $\mathbf w$, we have

Theorem 5. For simplicity let $\mathbf f_\tau=\mathbf f$, $\mathbf g_\tau=0$ and $\mathbf y_{0\tau}=\mathbf w_0$.

1. Let the assumptions on matrix coefficients in Theorem 4, part 1, hold. Then the difference $\mathbf r_\tau=\mathbf w-\mathbf y_\tau$ of the solutions of the Cauchy problems (2.1) and (3.1) satisfies the estimate

2. Let all assumptions on matrix coefficients of Theorem 4 hold. Then for ${1<\alpha<2}$ $\mathbf r_\tau$ also satisfies the estimate

In these estimates and similar estimates below it is automatically assumed that $\mathbf w_0$ and $\mathbf f$ belong to the spaces whose norms are applied.

Theorem 6. Let all assumptions of Theorem 1 for $q=1$ and of Theorem 3 hold, and let $\|A\|_{L^\infty(\Pi_T)}\leqslant N$ and $\|\operatorname{div} A_{\cdot j}\|_{L^\infty(\Pi_T)}\leqslant N$, $j=1,\dots,n$. Then the difference $\mathbf r_\tau=\mathbf w-\mathbf y_\tau$ of solutions of the Cauchy problems (2.1) and (3.10), (3.11) satisfies the estimate

In particular, if $\mathbf f_\tau=\mathbf f$, $\mathbf y_{0\tau}=\mathbf w_0$ and $\|\mathbf y_{1\tau}\|_{L^2(\mathbb{R}^n)} \leqslant C_1(N,T)\bigl(\|\mathbf w_0\|_{H^2(\mathbb{R}^n)}+\|\mathbf f\|_{W_{2,1}^{2,1}(\Pi_T)}\bigr)$ (for example, if $\mathbf y_{1\tau}=0$ or $\mathbf y_{1\tau}=(\partial_t\mathbf w)_0\equiv(\partial_t\mathbf w)|_{t=0}$), then the following simplified estimate is true:

Theorem 7. Let all assumptions on matrix coefficients of Theorem 6 hold, let $\mathbf y_{0\tau}=\overline{\sigma}^{(\tau)}\mathbf w_0$, and let $\mathbf f_\tau=\mathbf f$ and $\mathbf y_{1\tau}=0$ for simplicity. Then the difference $\mathbf r_\tau=\mathbf w-\mathbf y_\tau$ of solutions of the Cauchy problems (2.1) and (3.10), (3.11) satisfies the estimate

Theorem 8. 1. a) Let $\mathbf f\in L^{2,1}(\Pi_T)$ and $\mathbf w_0\in L^2(\mathbb{R}^n)$. Then the weak solution $\mathbf w$ of the Cauchy problem (2.1) satisfies the estimate

b) Let $p=1,2$. If $\mathbf f,\nabla^p\mathbf f\in L^{2,1}(\Pi_T)$ and $\mathbf w_0\in H^p(\mathbb{R}^n)$, then the strong solution $\mathbf w$ of the Cauchy problem (2.1) satisfies the estimates

If $p=2$ and, in addition, $\|\{\partial_t\mathbf B,\partial_tC\}\|_{L^\infty(0,T)}\leqslant N$ and $\partial_t\mathbf f\in L^{2,1}(\Pi_T)$, then

2. Assume that conditions (3.2) hold for $\mu=0$, and let $\mathbf f_\tau\in L^{2,1}(\Pi_T)$, $\mathbf g_\tau\in L^2(\Pi_T)$ and $\mathbf y_{0\tau}\in L^2(\mathbb{R}^n)$. Then the weak solution $\mathbf y=\mathbf y_\tau$ of the Cauchy problem (3.1) satisfies the estimate

3. Assume that conditions (3.2) for $\mu=0$, (3.13) for $\mu_1=0$ and (3.15) hold, let $\partial_tA_{ij}\in L^\infty(0,T)$, $A_{ij}=A_{ji}^T$, $i,j=1,\dots,n$, and also let $\mathbf f_\tau\in L^{2,1}(\Pi_T)$, $\mathbf y_{0\tau}\in H^1(\mathbb{R}^n)$ and $\mathbf y_{1\tau}\in L^2(\mathbb{R}^n)$. Then the weak solution $\mathbf y=\mathbf y_\tau$ of the Cauchy problem (3.10), (3.11) satisfies the estimate

Now, in averaging (2.10) we take a smooth kernel $\widehat{e}\in C^\infty(\mathbb{R})$ such that

Under the assumptions in parts 1, 2 and 3, applying the operator $\partial^{\mathbf k}$ to the Cauchy problems (5.9), (5.10) and (5.11) yields, respectively,

Taking the limit in the integral identities that are the definitions of weak solutions of the Cauchy problems (5.12), (5.13) and (5.14), (5.15) shows that $\mathbf w^{(\mathbf k)}$ and $\mathbf y^{(\mathbf k)}$ are weak solutions of the corresponding Cauchy problems

Now we deduce the estimates in part 1, b). Estimate (5.2) follows from (5.1) after we replace $\partial^{\mathbf k}$ by $\partial_i\partial^{\mathbf k}$ and $\partial_i\partial_j\partial^{\mathbf k}$ and sum over $i,j=1,\dots,n$. By virtue of the equation in (5.16) for $\mathbf w^{(\mathbf k)}=\partial^{\mathbf k}\mathbf w$, the derivatives $\partial_t^l\partial^{\mathbf k}\mathbf w$, $l=1,2$, can be expressed successively in terms of the $\nabla^p\partial^{\mathbf k}\mathbf w$, $p=0,l$, and the derivatives of $\mathbf f$ (like in parts c) and e) of the proof of Theorem 1); therefore, (5.3) and (5.5) follow from (5.1) and (5.2). Finally, for $p=2$, by virtue of the equation in (5.16) again, $\nabla\partial_t\mathbf w^{(\mathbf k)}=-B_i\nabla\partial_i\mathbf w^{(\mathbf k)}-C\nabla\mathbf w^{(\mathbf k)}+\nabla\partial^{\mathbf k}\mathbf f\in L^{2,1}(\Pi_T)$, and estimate (5.4) follows from (5.2). Theorem 8 is proved.

Theorem 9. Assume that the assumptions of Theorem 8, parts 1 and 2, hold except for the conditions on $\partial_t\mathbf f$ and $\partial_t\mathbf B,\partial_tC$, and that $\|A\|_{L^\infty(0,T)}\leqslant N$. Let $\mathbf f_\tau=\mathbf f$, $\mathbf g_\tau=0$, $\mathbf y_{0\tau}=\mathbf w_0$ and $|\mathbf k|_1\geqslant 0$. Then the difference $\mathbf r_\tau=\mathbf w-\mathbf y_\tau$ of solutions of the Cauchy problems (2.1) and (3.1) satisfies the estimates

Theorem 10. Let all assumptions of Theorem 8, part 1 for $q=1$ and part 3, hold and let $\|A\|_{L^\infty(0,T)}\leqslant N$. Let $\mathbf f_\tau=\mathbf f$, $\mathbf y_{0\tau}=\mathbf w_0$, $\mathbf y_{1\tau}=0$ and $|\mathbf k|_1\geqslant 0$. Then the difference $\mathbf r_\tau=\mathbf w-\mathbf y_\tau$ of solutions of the Cauchy problems (2.1) and (3.10), (3.11) satisfies the estimate

For $\mathbf y_{0\tau}=\overline{\sigma}^{(\tau)}\mathbf w_0$ (instead of $\mathbf y_{0\tau}=\mathbf w_0$) and $0<\alpha\leqslant 2$, it is also true that

Lemma 1. The symmetry and nonnegative definiteness properties

It is straightforward to see in view of the equality $1/\gamma+1/\gamma_*=1$ that the following pointwise formula is true:

The left-hand inequality in (6.9) is ensured by (6.8), where the following estimate integrated over $\mathbb{R}^n$ is taken into account:

The right-hand inequality in (6.9) is obvious. The lemma is proved.

Lemma 2. The following relations hold:

We introduce the matrices

This formula does not imply an inequality of type (3.14) for $\delta=0$. Nevertheless, inequality (6.10) of type (3.13) for $\delta=0$ is valid. To derive it, like in the difference case (see [17]), for $\mathbf z=(\rho,\mathbf u,\varepsilon)\in H^1(\mathbb{R}^n)$ and $\mathbf u=(u_1,\dots,u_n)$ we set $\mathbf v_i=\partial_i\mathbf z$ and integrate the last formula over $\mathbb{R}^n$. For all $k\neq l$, integrating twice by parts we obtain $(\partial_ku_l,\partial_lu_k)_{\mathbb{R}^n} =(\partial_lu_l,\partial_ku_k)_{\mathbb{R}^n}$ (the arising additional condition $\partial_k\partial_lu_k\in L^2(\mathbb{R}^n)$ is eliminated because $H^2(\mathbb{R}^n)$ is dense in $H^1(\mathbb{R}^n)$). Owing to this formula, $(\partial_iu_j,\partial_ju_i)_{\mathbb{R}^n}=\|\operatorname{div}\mathbf u\|_{\mathbb{R}^n}^2$, which leads to (6.10). The lemma is proved.

Theorem 11. 1. Theorem 1 and Theorem 8, part 1, for $c_0=0$ hold for the Cauchy problem for the linearized system of gas dynamic equations (6.6).

2. Let $\widehat{\alpha}_s>0$ and $\widehat{\alpha}_P>0$. Then for the Cauchy problems for the linearized quasi-gasdynamic systems (6.4), (6.5), property (3.2) holds for $\nu=c_*^2\delta_1$ and $\mu=0$ and property (3.13) holds for $\delta=\mu_1=0$. Hence Theorem 2 and Theorem 8, part 2, for $\overline{c}_0=c_0=0$ hold in the case $\ell=0$, while Theorem 3 and Theorem 8, part 3, for $\overline{c}_1=c_1=0$ hold in the case $\ell=1$.

The difference $\mathbf r_\tau=\mathbf w-\widetilde{\mathbf z}$ of solutions of the Cauchy problems for the linearized system of gas dynamic equations (6.6) and the linearized quasi-gasdynamic systems (6.4), (6.5) satisfies the estimates in Theorems 4 and 9 for $c_0=0$ in the case $\ell=0$ and in Theorems 6 and 10 for $c_0=c_1=0$ in the case $\ell=1$.