Let u and v be two weak solutions, (u x,0)=v x,0)).
Let u and v be two weak solutions with the different initial values (u x,0),v x,0)), respectively.
Let u and v be two weak solutions of equation (1.1) with the different initial values (u x,0)=v x,0)), respectively.
Let u and v be two weak solutions of equation (1.1) with the initial values (u x,0)), (v x,0)), respectively.
Let u and v be two weak solutions of equation (1.5) with the same boundary value condition (2.5) and with the initial values (u x,0)), (v x,0)), respectively.
Then frac{d}{dt} Vert {u - v} Vert _{L^{1}(Omega)} leq 0. It implies that int_{Omega}biglvert u x,t) - v x,t) bigrvert,dx leq int_{Omega}biglvert u_{0}(x) - v_{0}(x) bigrvert,dx,quad forall t in[0,T). □. Let u and v be two weak solutions of equation (1.5) with the initial values (u x,0)), (v x,0)), respectively.
Let (u_{1}, u_{2}in W_{1}(M,T)) be two different weak solutions of Prob.
Let and be two probabilistic weak solutions of (1.1), (1.2) starting with the same initial condition and defined on the same stochastic basis with the same Wiener process.
Assume that (AF)–(ASFG) hold and let and be two probabilistic weak solutions of (1.1), (1.3) starting with the same initial condition and defined on the same stochastic basis.
Assume that (HYP 1)–(HYP 4) hold and let and be two probabilistic weak solutions of (1.1), (1.2) starting with the same initial condition and defined on the same stochastic basis with the same Winer process.
Here are four weak counterexamples.
