Sentence examples for maximal weak solution from inspiring English sources
Then, there exist a minimal weak solution and a maximal weak solution of (2.3) satisfying (2.10).
Similar to the proof Theorem 1.1 in [10] and the proof of Theorem 7.5.1 in [23], it is not difficult to get from Theorem 2.1 that is the maximal weak solution and the minimal solution of (2.3), which satisfies (2.10) and.
Let u 0 ( x ), u 1 ( x ) ∈ H 2 be given and u ( x, t ) be a weak solution of problem (1.1 - 1.2 1.1 - 1.2imal existence time interval [ 0, T 0 ), T 0 ≤ + ∞. Assume that the initial data satisfy (3.1).
(Gradient sharp maximal estimate [40]) Let u ∈ W 1, p be a weak solution to (13) under the assumptions (16).
Then we have the following: (Intermediate maximal estimates [40]) Let u ∈ W 1, p be a weak solution to (13) under the assumptions (16).
(Uniform maximal-potential estimates) Let u ∈ W 1, p be a weak solution to the Eq. (13) under assumptions (16).
Proof of Theorem 3.1 From Theorem 2.1 there exists a unique local solution of problem (1.1 - 1.2 1.1 - 1.2on a maximal time interval [ 0, T 0 ). Let u ( t ) be the weak solution of problem (1.1)-(1.2) with E ( 0 ) > 0, u 0 ∈ W andefined. Then fron Lemma 3.3 we have u ( x, t ) ∈ W, namaximal ( u ( time) > ∥ u t ∥ 2 + ∥ u x x t ∥ 2 for t ∈ [ 0, T 0 ). (3.10).
Let (u t,x)) be any weak solution of problem (5) with (phiin W_{1}), T be the maximal existence time of (u t,x)).
Note that the variational inequalities (1.1) are concerned with the Lorentz space with variable exponent powers (p(x)) of the gradient of weak solution, so that the techniques from harmonic analysis, like the Calderón-Zygmund operator, the maximal function operator and the sharp maximal function operator, might not be suitable for our estimates.
Definition 2.1 (Weak solution).
