A Class Of Non-Newtonian Fluids With External Force And Vacuum Under No Compatibility Conditions

We consider the one-dimensional equations of compressible non-Newtonian fluids which read as follows:where t>0,x ? R,?0>0,p>2,the unknown functions ?= ??x,t?= u = u?x,t?and ????= A???A>0,?>1?denote the density,the velocity and the pressure,separately.Without loss of generality,we set A = 1.We consider the Cauchy problem with??,u?vanishing at infinity.For given initial functions,we require that The motion of the fluid is driven by an external force f?t,x,u?.satisfying where C1,C2,C3,C4,C5,C6 are positive constant.Assume H₁?t,x?? 0,H₂?t,x?? 0,H₃?t,x?? 0 for all?t,x??(0,1]x?-?,+??,satisfying where q ? p is a positive constant and C1,C2,C3,C4,C5,C6 are positive constant.We get the following results:Theorem 1 Assuming that f?t,x,y?as?3?-?4?and assume that the initial data??0,u0?satisfies where?t,x??(0,T0]× R.Further,for constant p>2,q ? p.assume that ?0 also satisfies where and ?0 is a positive constant.Then there exists a positive time T0?T0 ? 1?such that the problem?1?-?2?has a unique strong solution??,u?on R ×(0,T0]satisfying Moreover,for some constantWe begin with the following standard energy estimate to initial-boundary value problem?1?-?2??approximation equation?.The introduction of ? are defined as follows After that,by computation,there exist positive constants T0 and M such that andFinally,we use truncation technique and standard arguments to obtain the local existence and uniqueness of strong solutions.For the Cauchy problem?1?-?2?,it is still open even for the local existence of strong solutions under no compatibility conditions when the far field density is vacuum.In the absence of compatibility condition,in addition to this article,so far,the literature on the local strong solutions of the Cauchy problem for such non Newtonian fluids has not been found.Moreover,The paper[20]motivated our study.Compared with[1],the advantages of this paper is not need compatibility conditions.We get the conclusion in R and overcome the difficulties caused by the external force.And compared with[20],the system?1?is with strong nonlinearity,so we are facing another difficulty.The authors in[20]bounded the LP?R2?-norm of u just in the terms of ??1/2u?L²?R²?and?ux?L²?R²?by Hardy type and Poincare type inequalities.In a similar way,we bounded the Lk?R??k>p?-norm of u just in the terms of ??1/2u?L2?R?and ?ux?Lp?R?.However,the application of a Sobolev embedding theorem in R is very different from R2.The reason is that the results of compact embedding in R are much less than that in R2.For this,the existence of local strong solutions is proved by the truncation technique and the standard arguments,and the uniqueness results are obtained.