Full Text
P. E. SOBOLEVSKII
COERCIVITY INEQUALITIES FOR ABSTRACT PARABOLIC EQUATIONS
(Presented by Academician I. N. Vekua, 4 II 1964)
1. The problem considered is
\[ \frac{dv}{dt}+Av=f(t), \qquad v(0)=v_0 \tag{1} \]
in a Banach space \(E\). Here \(v(t)\) and \(f(t)\) are the unknown and the given functions, defined on \([0,T]\), with values in \(E\); \(dv/dt\) is the derivative, understood as the limit in the norm of \(E\) of the corresponding difference quotient; \((-A)\) is the infinitesimal operator of the strongly continuous semigroup \(\exp\{-tA\}\).
As is known, for sufficiently smooth \(v_0\) and \(f(t)\), problem (1) has a unique continuously differentiable solution \(v(t)\), and
\[ v(t)=\exp\{-tA\}v_0+\int_0^t \exp\{-(t-s)A\}f(s)\,ds . \tag{2} \]
In the theory of semigroups it is proved that the operator \(A+\lambda I\) has a bounded inverse if \(\operatorname{Re}\lambda\geqslant\sigma_0\) and \(\sigma_0\) is a sufficiently large positive number. If
\[ \|(A+\lambda I)^{-1}\|_{E\to E}\leqslant C(|\lambda|+1)^{-1} \qquad (\operatorname{Re}\lambda\geqslant\sigma_0), \tag{3} \]
then we shall call \(A\) strongly positive. It is known that in this case the semigroup is continuously differentiable for \(t>0\), and
\[ \|A\exp\{-tA\}\|_{E\to E}\leqslant C(T)t^{-1} \qquad (0<t\leqslant T). \tag{4} \]
2. The totality of all continuous on \([0,T]\) functions \(w(t)\) with values in \(E\), satisfying, for some \(\alpha\in(0,1)\), the condition
\[
t^\alpha(\Delta t)^{-\alpha}\,\|w(t+\Delta t)-w(t)\|_E\leqslant C(\alpha)
\qquad (0<t<t+\Delta t\leqslant T),
\]
will be denoted by \(C_0^\alpha(T)\). This is a Banach space with norm
\[ \|w\|_{C_0^\alpha(T)} = \max_{0\leqslant t\leqslant T}\|w(t)\|_E + \sup_{0<t<t+\Delta t\leqslant T} t^\alpha(\Delta t)^{-\alpha}\|w(t+\Delta t)-w(t)\|_E . \tag{5} \]
If, for every \(v_0\in D(A)\) and every \(f(t)\in C_0^\alpha(T)\), there exists such a solution of problem (1) that
\[ \left\|\frac{dv}{dt}\right\|_{C_0^\alpha(T)} + \|Av\|_{C_0^\alpha(T)} \leqslant C(\alpha,T)\bigl(\|f\|_{C_0^\alpha(T)}+\|Av_0\|_E\bigr), \tag{6} \]
then we shall say that, for problem (1), coercivity holds in \(C_0^\alpha(T)\).
Theorem 1. In order that coercivity hold for problem (1) in \(C_0^\alpha(T)\), it is necessary and sufficient that the operator \(A\) be strongly positive in \(E\).
Necessity. Let \(f_0\) be an arbitrary element of \(E\), and let \(v_0\) be the solution of the equation \(\lambda v_0 + A v_0 = f_0\), which exists for \(\operatorname{Re}\lambda \geqslant \sigma_0\). The function \(v(t)=\exp\{t\lambda\}v_0\) is a solution of problem (1) for \(f(t)=\exp\{t\lambda\}f_0\). Using (6), we obtain (3).
To prove sufficiency, one must use estimate (4) and apply the ordinary theory of singular integrals to the singular integral
\[ A\int_0^t \exp\{-(t-s)A\} f(s)\,ds \tag{7} \]
The inequality (6) makes it possible to study a problem more general than (1),
\[ \frac{dv}{dt}A(t)v=f(t), \qquad v(0)=v_0, \tag{8} \]
where \(A(t)\), for each \(t\in[0,T]\), is a strongly positive operator whose domain of definition does not depend on \(t\). Let \(A(0)=A\), and let \(A+\sigma_0 I\) have an inverse. Introduce the operator \(B(t)=A(t)(A+\sigma_0 I)^{-1}\) and, for example, suppose that the operator-valued function \(B(t)\) satisfies on \([0,T]\) a Hölder condition with exponent \(\alpha\). Then for solutions of problem (8) coercivity holds in \(C_0^\alpha(T)\).
- In the theory of problem (1) on the finite segment \([0,T]\), without loss of generality one may assume that \(\sigma_0=0\). Then for any \(\alpha\in[0,1]\) and, for example, for any \(v_0\in D(A)\), the number
\[ |v_0|_\alpha = \left( \int_0^\infty \|A\exp\{-tA\}v_0\|_E^{\frac{1}{1-\alpha}}\,dt \right) \tag{9} \]
is finite.
The set of all those \(v_0\) for which \(|v_0|_\alpha<\infty\) forms a linear set \(E_\alpha\). If \(A\) is strongly positive, then \(E_\alpha\) is a Banach space with norm (9). In (¹) fractional powers \(A^\alpha\) of a broad class of operators \(A\), acting in arbitrary Banach spaces, were introduced. Such operators include, in particular, strongly positive operators. The domain of definition \(D_\alpha\) of the operator \(A^\alpha\) is a Banach space with norm \(\|v_0\|_\alpha=\|A^\alpha v_0\|_E\).
Theorem 2. Let \(A\) be strongly positive, \(\alpha\in(0,1)\), and \(\varepsilon\in(0,\alpha)\). Then:
\[ \text{if } v_0\in D_\alpha,\quad \text{then } v_0\in E_{\alpha-\varepsilon} \text{ and } |v_0|_{\alpha-\varepsilon}\leqslant C(\alpha,\varepsilon)\|v_0\|_\alpha; \]
\[ \text{if } v_0\in E_\alpha,\quad \text{then } v_0\in D_{\alpha-\varepsilon} \text{ and } \|v_0\|_{\alpha-\varepsilon}\leqslant C(\alpha,\varepsilon)|v_0|_\alpha. \]
The question arises for which spaces \(E\) and operators \(A\) Theorem 2 remains valid also for \(\varepsilon=0\).
If \(E\) is a Hilbert space \(H\), and \(A\) is a positive definite self-adjoint operator in it, then for \(\alpha\geqslant 1/2\), if \(v_0\in D_\alpha\), then \(v_0\in H_\alpha\) and \(|v_0|_\alpha\leqslant C(\alpha)\|v_0\|_\alpha\). M. A. Krasnosel’skii showed that for \(\alpha\in(0,1/2)\) the last assertion is, generally speaking, false. Finally, it is easy to see that \(D_{1/2}=H_{1/2}\).
- By \(B_p(T)\) (\(p\geqslant1\)) we denote the Bochner space of strongly measurable functions \(w(t)\) on \([0,T]\) with values in \(E\), whose norm is summable to the power \(p\). The norm in \(B_p(T)\) is defined, as is known, by the formula
\[ \|w\|_p^T = \left( \int_0^T \|w(t)\|_E^p\,dt \right)^{1/p}. \tag{10} \]
If for any \(f(t)\in B_p(T)\) and any \(v_0\in E_{1/q}\), where \(1/p+1/q=1\), there exists such an absolutely continuous solution \(v(t)\) of problem (1) that \(dv/dt, Av\in B_p(T)\), and the function \(v(t)\) is continuous on \([0,T]\) in the norm of the space
\(E_{1/q}\), and for every \(t \in [0,T]\) the inequality
\[ \left\| \frac{d\bar v}{dt} \right\|_{-p} + \left\| A\bar v \right\|_{p}^{t} + |v(t)|_{1/q} \leq C(p,T)\left(\|f\|_{-p}^{t}+|v_0|_{1/q}\right), \tag{11} \]
holds, then we shall say that coercivity holds for problem (1) in \(B_p(T)\).
Theorem 3*. In order that coercivity in \(B_p(T)\) hold for problem (1), it is necessary that the operator \(A\) be strongly positive in \(E\). This condition is also sufficient if \(1<p<\infty\) and if
\[ \text{coercivity holds for some } r \in (1,\infty). \tag{*} \]
The necessity is proved in the same way as in Theorem 1. The proof of sufficiency is based on a theorem from the theory of singular integrals in \(B_p(T)\) \((^2)\) and on the well-known Hardy inequality.
If \(E\) is \(H\), then with the aid of the Fourier transform it is shown that coercivity holds in \(B_2(T)\). Therefore, by Theorem 3, coercivity holds also in any \(B_p(T)\).
Inequality (11) makes it possible to show that coercivity in \(B_p(T)\) also holds for the solutions of problem (8), if \(B(t)\) (see item 2) has only discontinuities of the first kind.
- Let \(\Omega\) be a bounded domain of \(n\)-dimensional space with a sufficiently smooth boundary. In the cylinder \(Q=\Omega\times[0,T]\) consider the parabolic system
\[ \frac{\partial \bar v}{\partial t} + A\left(x,\frac{\partial}{\partial x}\right)\bar v = \bar f(t,x) \tag{12} \]
with an elliptic differential operator in partial derivatives of order \(2l\). For solutions of system (12) satisfying homogeneous boundary conditions, for example in \((^3)\), coercivity inequalities have been established in Schauder norms and in the norms \(L_p(Q)\). In these inequalities the norms with respect to \(t\) and with respect to the spatial variables are the same.
The boundary-value problem for system (12) can be regarded as the Cauchy problem (1) in various functional spaces. The application of Theorems 1 and 3 to the results of \((^3)\) makes it possible to obtain a series of new coercivity inequalities with different norms with respect to \(t\) and with respect to the spatial variables. For lack of space we do not write out these rather cumbersome inequalities.
In \((^4)\), and then also in \((^3)\), the first boundary-value problem for the linearized Navier—Stokes system in the cylinder \(Q\) was considered, and coercivity inequalities were established for the solutions of this system. Theorems 1 and 3 here also lead to new coercivity inequalities.
A second consequence of Theorems 1 and 3 for parabolic systems and the Navier—Stokes system is that the corresponding stationary systems generate strongly positive operators. This, in particular, makes it possible to develop an \(L_p\)-theory for the nonlinear nonstationary Navier—Stokes system and to prove that in the \(n\)-dimensional case a local theorem of existence and uniqueness of the solution of the first boundary-value problem for such a system is valid for any solenoidal initial velocity \(v_0\) from \(L_n(\Omega)\).
- Coercivity inequalities also hold in norms with derivatives. We formulate here only one result.
Let \(w(t)\) be a function continuous on \([0,T]\) with values in \(E\). Suppose that for \(t>0\) this function is \(k\) times continuously differentiable and its derivative \(w^{(k)}(t)\) satisfies a Hölder condition with exponent \(\alpha\). Finally, suppose that
\[ \sum_{i=0}^{k} \sup_{0<t\leq T} t^i\|w^{(i)}(t)\|_E + \sup_{0<t<t+\Delta t\leq T} t^{k+\alpha}(\Delta t)^{-\alpha} \|w^{(k)}(t+\Delta t)-w^{(k)}(t)\|_E <\infty. \tag{13} \]
\[ \text{* Apparently, Theorem 3 is also true without condition (*).} \]
The totality of all such functions forms a Banach space \(C_0^{\alpha+k}(T)\) with norm \(\|w\|_{C_0^{\alpha+k}(T)}\), defined by the left-hand side of inequality (13).
We shall say that coercivity in \(C_0^{\alpha+k}(T)\) holds for the solutions of problem (1) if, for any \(v_0\in D(A)\) and \(f(t)\in C_0^{\alpha+k}(T)\), there exists a solution of problem (1) such that \(dv/dt,\ Av\in C_0^{\alpha+k}(T)\), and the inequality
\[ \left\|\frac{dv}{dt}\right\|_{C_0^{\alpha+k}(T)} +\|Av\|_{C_0^{\alpha+k}(T)} \leq C(\alpha+k,T)\left(\|f\|_{C_0^{\alpha+k}(T)}+\|Av_0\|_E\right) \tag{14} \]
holds.
A generalization of Theorem 1 is
Theorem 4. In order that coercivity in \(C_0^{\alpha+k}(T)\) hold for the solutions of (1), it is necessary and sufficient that the operator \(A\) be strongly positive in \(E\).
This theorem makes it possible to investigate problem (8) and to show that coercivity in \(C_0^{\alpha+k}(T)\) holds for its solutions if, for example, the operator-function \(B(t)\) is \(k\) times continuously differentiable on \([0,T]\) and its derivative \(B^{(k)}(t)\) satisfies a Hölder condition with exponent \(\alpha\).
In conclusion we merely note that analogous assertions also hold in the spaces \(B_p(T)\).
Received
4 II 1964
REFERENCES
\(^{1}\) M. A. Krasnosel’skii, P. E. Sobolevskii, DAN, 129, No. 3 (1959).
\(^{2}\) A. Benedek, A. P. Calderon, R. Panzone, Proc. Nat. Acad. Sci. USA, 48, No. 3 (1962).
\(^{3}\) V. A. Solonnikov, Reports at the joint Soviet-American symposium, Novosibirsk, 1963.
\(^{4}\) V. I. Yudovich, DAN, 130, 1214 (1960).