Generalized solutions describing singularity interaction
The author considers initial value problems connected with nonlinear differential equations of the second degree, say \(L[u]= 0\), with initial data of the form \(u(x,0)= u_0+u_1\theta (-x+a_1)+u_2 \theta(-x+ a_2)\), where \(u_1,u_2,a_1\) and \(a_2\) stand for given real constants while \(\theta (x)\) is the Heaviside unit step function, and gives a method to find an approximate weak solution valid in the asymptotic sense when certain parameter \(\varepsilon\) tends to zero. This approximation requires to replace \(u(x,t)\) by a function \(u_\varepsilon(x,t)\) such that \(\langle L[u_\varepsilon], \eta\rangle=O (\varepsilon)\) for any \(\eta(x)\in C^\infty_0\). The method is based on the identity \[ \omega_1\left({x-a_1\over \varepsilon}\right) \omega_2\left({x-a_2 \over\varepsilon} \right)=\theta (x-a_1)B_1\left({\Delta a\over \varepsilon} \right)+\theta (x-a_2)B_2\left({\Delta a\over \varepsilon} \right)+ O_D (\varepsilon), \] where \(\omega_j(z)\in C^\infty\) stands for any function such that its derivative belongs to the Schwartz space and \(\lim_{z\to-\infty} \omega_j =0\), \(\lim_{z\to\infty} \omega_j=1\), while \(B_j(z)\) refer to two functions defined in terms of \(\omega_1\) and \(\omega_2\). As to \(O_D (\varepsilon)\), it denotes any distribution, say \(f\), such that \(\langle f,\eta \rangle= O(\varepsilon)\). The method is applied to the classical Hopf equation for which \(L[u]=u_t +(u^2)_x\). An extension of the method to the case when \(x\) belongs to a two-dimensional space is also shown on a particular example. Remark that this second application contains some misprints.
scientific article; zbMATH DE number 1780366