Well-posedness And Blow-up Criterion For The One-dimensional Fractal Burgers Equation

This paper considers the Cauchy problem for a one-dimensional fractal Burgers equa-tion, i.e. ut+(-△)αu+uxu=0. First, we establish a corresponding bilinear estimate in the functional space Xs(R), then by the contraction mapping principle we prove the exis-tence and uniqueness of local Lei-Lin solution to the Cauchy problem of the fractal Burgers equation with the initial value u0∈X1-2α(R). Finally, we obtain the global existence of the small initial value solution and give a blow-up criterion of the local solution.In chapter two, first, we introduce a functional space Xs(R) to be the working space, and estimate nonlinear term in the space-time space. Then we construct a mapping by using mild solution form and a suitable space-time space, and prove that the mapping is a contraction mapping in the subspace of the space-time space. Finally, by the contrac-tion mapping principle, we obtain the existence and uniqueness of the solution in the sub-space of the space-time space to the Cauchy problem for any initial value. For proving the existence, we suppose that both u1(t,x) and u2(t,x) belong to the space-time space and are the solutions of the Cauchy problem, we can give u1(0,x)= u2(0,x). Then we let u(t,x)= u1(t, x)-u2(t,x), and put the new u(t,x) into the equation. We take Fourier transform for the spatial variable and use the method of regularization on time, then we can prove that u= 0 in [0, T] by the Gronwall’s lemma, so the existence of the Cauchy problem in the space-time space can be proved.In chapter three, on the basis of the existence and uniqueness of the solution in the subspace of the space-time space, we take Fourier transform for the spatial variable. Then we can obtain the global existence of the small initial value solution by the the method of regularization on time. Finally, using the normal skills of the extension so-lution in ordinary differential equation, we establish a blow-up criterion of the local solution.