In this paper,we study the global existence of BV solutions of the initial value problem for the isentropic p-system,where the state equation of the gas is given by P=Av^(-γ).Forγ>1,the general existence result f...In this paper,we study the global existence of BV solutions of the initial value problem for the isentropic p-system,where the state equation of the gas is given by P=Av^(-γ).Forγ>1,the general existence result for large initial data has not been obtained.By using the Glimm scheme,Nishida,Smoller and Diperna successively obtained the global existence results for(γ-1)TV(v_(0)(x),u_(0)(x))being small.In the present paper,by adopting a rescaling technique,we improve these results and obtain the global existence result under the condition that(γ-1)^(γ+1)(TV(v_(0)(x)))~(γ-1)(TV(u_(0)(x)))^(2) is small,which implies that,for fixedγ>1,either TV(v_(0)(x))or TV(u_(0)(x))can be arbitrarily large.展开更多
We are concerned with the uniqueness of solutions of the Cauchy problemand a(s),b(s) are appropriately smooth.Since a(s) is allowed to have zero points, we call them points of degeneracy of (1), the equation (1) does ...We are concerned with the uniqueness of solutions of the Cauchy problemand a(s),b(s) are appropriately smooth.Since a(s) is allowed to have zero points, we call them points of degeneracy of (1), the equation (1) does not admit classical solutions in general. The solutions of (1) even might be discontinuous, whenever the set E = {s : a(s) = 0} includes interior points.Equations with degeneracy arise from a wide variety of diffusive processes in nature展开更多
In this paper,we propose a new non-local diffusion equation for noise removal,which is derived from the classical Perona-Malik equation(PM equation)and the regularized PM equation.Using the convolution of the image gr...In this paper,we propose a new non-local diffusion equation for noise removal,which is derived from the classical Perona-Malik equation(PM equation)and the regularized PM equation.Using the convolution of the image gradient and the gradient,we propose a new diffusion coefficient.Due to the use of the convolution,the diffusion coefficient is non-local.However,the solution of the new diffusion equation may be discontinuous and belong to the bounded variation space(BV space).By virtue of Young measure method,the existence of a BV solution to the new non-local diffusion equation is established.Experimental results illustrate that the new method has some non-local performance and performs better than the original PM and other methods.展开更多
基金partially the NSFC(11671193)Fangqi Chen was partially the NSFC(12172166,11872201)。
文摘In this paper,we study the global existence of BV solutions of the initial value problem for the isentropic p-system,where the state equation of the gas is given by P=Av^(-γ).Forγ>1,the general existence result for large initial data has not been obtained.By using the Glimm scheme,Nishida,Smoller and Diperna successively obtained the global existence results for(γ-1)TV(v_(0)(x),u_(0)(x))being small.In the present paper,by adopting a rescaling technique,we improve these results and obtain the global existence result under the condition that(γ-1)^(γ+1)(TV(v_(0)(x)))~(γ-1)(TV(u_(0)(x)))^(2) is small,which implies that,for fixedγ>1,either TV(v_(0)(x))or TV(u_(0)(x))can be arbitrarily large.
基金Partially supported by NSF (19631050) of China, partially supported by the grant of Ministry of Science and Technologies of China, and partially supported by the Outstanding Young Fundation (19125107) of China.
文摘We are concerned with the uniqueness of solutions of the Cauchy problemand a(s),b(s) are appropriately smooth.Since a(s) is allowed to have zero points, we call them points of degeneracy of (1), the equation (1) does not admit classical solutions in general. The solutions of (1) even might be discontinuous, whenever the set E = {s : a(s) = 0} includes interior points.Equations with degeneracy arise from a wide variety of diffusive processes in nature
基金partially supported by the National Natural Science Foundation of China(11971131,12171123,11871133,11671111,U1637208,61873071,51476047)the Guangdong Basic and Applied Basic Research Foundation(2020B1515310006)+1 种基金the Natural Sciences Foundation of Heilongjiang Province(LH2021A011)China Postdoctoral Science Foundation(2020M670893).
文摘In this paper,we propose a new non-local diffusion equation for noise removal,which is derived from the classical Perona-Malik equation(PM equation)and the regularized PM equation.Using the convolution of the image gradient and the gradient,we propose a new diffusion coefficient.Due to the use of the convolution,the diffusion coefficient is non-local.However,the solution of the new diffusion equation may be discontinuous and belong to the bounded variation space(BV space).By virtue of Young measure method,the existence of a BV solution to the new non-local diffusion equation is established.Experimental results illustrate that the new method has some non-local performance and performs better than the original PM and other methods.
