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Existence, uniqueness and regularity for Kruzkov’s solutions of the Burger-Carleman’s system. (English) Zbl 0725.35061

The authors prove existence and uniqueness of Kruzkov’s weak solution (u(t),v(t)) to the initial value problem for the Burger-Carleman’s system on \([0,\infty)\times {\mathbb{R}}:\) \[ u_ t+(u^ 2/2)_ x+u^ 2-v^ 2=0;\quad v_ t-(v^ 2/2)_ x+v^ 2-u^ 2=0 \] with the initial data \((u_ 0,v_ 0)\in L^ 1({\mathbb{R}})^ 2_+\). They reduce the problem to solve an abstract Cauchy problem: \[ (*)\quad dU/dt+(A+B)U=0,\quad U(0)=(u_ 0,v_ 0)\in L^ 1({\mathbb{R}})^ 2_+ \] in the Banach lattice \(E=L^ 1({\mathbb{R}})^ 2\) where \(AU=((u^ 2/2)_ x,-(v^ 2/2)_ x)\), \(BU=(u^ 2-v^ 2,v^ 2-u^ 2)\) for \(U=(u,v)'.\)
To solve (*) they apply the Crandall-Liggett’s theorem in the nonlinear semigroup theory by showing that \(A+B\) is T-accretive in E and that \(R[I+\lambda (A+B)]=L^ 1({\mathbb{R}})^ 2_+\) for all \(\lambda >0\). Then in making use of \(L^ 1-L^{\infty}\) regularizing effect, they show that the semigroup solution is actually the Kruzkov’s solution.
Reviewer: T.Kakita (Tokyo)

MSC:

35L80 Degenerate hyperbolic equations
47H20 Semigroups of nonlinear operators
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References:

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