In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma...In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-′qF(q) + L(′F(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T~*. Meanwhile, we can give the equation of the shock and an explicit value of T~*.展开更多
In this paper,we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space.First,we analyze some properties of the shock and rarefaction wave curves in the Riemann invaria...In this paper,we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space.First,we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane.Based on these properties,we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme,and prove that there exists an entropy solution V(x,t)which belongs to L^(∞)∩BV_(loc)(R×R_(+)).展开更多
In this article, we give the existence of global L^(∞)bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of Le Roux type. The main difficulty lies in establishing some compa...In this article, we give the existence of global L^(∞)bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of Le Roux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2×2 to n×n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1= 0} is another difficulty.We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.展开更多
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
Transmembrane water exchange(TWE)including transcytolemmal water exchange and transvascular water exchange is involved in many in vivo measurements and makes different contributions to the measuring results.In this st...Transmembrane water exchange(TWE)including transcytolemmal water exchange and transvascular water exchange is involved in many in vivo measurements and makes different contributions to the measuring results.In this study,we focus on the potential influence of TWE on the cell density parameter,intracellular water mole fraction pi,derived by dynamic contrast enhanced-magnetic resonance imaging(DCE-MRI)which has been reported as a technique to characterize perfusion and vascularization of tissues,but its accuracy in measuring cell density(or interstitial space)has been questioned.Sixteen patients with glioblastoma multiforme(GBM)were enrolled since GBM shows strong intratumor heterogeneity in both cell density and TWE.All the subjects were collected with DCE-MRI and apparent diffusion coefficient(ADC)map.The latter was considered as a valid surrogate of cell density.Extended Tofts(eTofts)model considering TWE as infinitely large variables and shutter-speed model(SSM)considering TWE as finite ones were used to fit DCE-MRI data.Monte Carlo(MC)and finite difference(FD)methods were used to simulate the influence of TWE on DCE-MRI-derived pi and ADC,respectively.The eTofts model shows a significant overestimation of pi in comparison with SSM in GBM(P<0.001),which is in accordance with MC simulations,and this overestimation shows dependence on the intra-to-extracellular water exchange rate constant(kio).Significant negative correlations between ADC and SSM-derived pi were found in both voxel-wise analyses(t-test P<0.001,average r=-0.74)and inter-subject comparisons(r=-0.63,P=0.009).But no consistent voxel-wise correlations(P>0.05)and a weaker inter-subject negative correlation(r=-0.56,P=0.02)were found between ADC and eTofts-derived pi.Further experimental and FD results revealed that kio made a limited contribution to ADC values in the physiological kio range in GBM,supporting ADC as a valid biomarker of cell density.These results suggest that the DCE-MRI pharmacokinetic shutter-speed model could significantly improve its accuracy in cell density estimation because of the considering transmembrane water exchange.展开更多
基金supported in part by NSFC(11671193)the Fundamental Research Funds for the Central Universities(NE2015005)supported in part by NSFC(11271182 and 11501290)
文摘In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x = 0. First, we prove a simple but useful property of Lax-Oleinik formula(Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-′qF(q) + L(′F(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T~*. Meanwhile, we can give the equation of the shock and an explicit value of T~*.
基金supported by NSFC(11671193)the Fundamental Research Funds for the Central Universities NE2015005。
文摘In this paper,we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space.First,we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane.Based on these properties,we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme,and prove that there exists an entropy solution V(x,t)which belongs to L^(∞)∩BV_(loc)(R×R_(+)).
基金supported by the National Science Foundation of China(11572148,11671193)the National Research Foundation for the Doctoral Program of Higher Education of China(20133218110025)
文摘In this article, we give the existence of global L^(∞)bounded entropy solutions to the Cauchy problem of a generalized n × n hyperbolic system of Le Roux type. The main difficulty lies in establishing some compactness estimates of the viscosity solutions because the system has been generalized from 2×2 to n×n and more linearly degenerate characteristic fields emerged, and the emergence of singularity in the region {v1= 0} is another difficulty.We obtain the existence of the global weak solutions using the compensated compactness method coupled with the construction of entropy-entropy flux and BV estimates on viscous solutions.
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the National Natural Science Foundation of China(NSFC)(Grant No.81873894,Grant No.21922410)Natural Science Foundation of Zhejiang Province,China(Grant No.LR20H180001,Grant No.LR19B050001)Taishan Scholars Program(No.tsqn20161070).
文摘Transmembrane water exchange(TWE)including transcytolemmal water exchange and transvascular water exchange is involved in many in vivo measurements and makes different contributions to the measuring results.In this study,we focus on the potential influence of TWE on the cell density parameter,intracellular water mole fraction pi,derived by dynamic contrast enhanced-magnetic resonance imaging(DCE-MRI)which has been reported as a technique to characterize perfusion and vascularization of tissues,but its accuracy in measuring cell density(or interstitial space)has been questioned.Sixteen patients with glioblastoma multiforme(GBM)were enrolled since GBM shows strong intratumor heterogeneity in both cell density and TWE.All the subjects were collected with DCE-MRI and apparent diffusion coefficient(ADC)map.The latter was considered as a valid surrogate of cell density.Extended Tofts(eTofts)model considering TWE as infinitely large variables and shutter-speed model(SSM)considering TWE as finite ones were used to fit DCE-MRI data.Monte Carlo(MC)and finite difference(FD)methods were used to simulate the influence of TWE on DCE-MRI-derived pi and ADC,respectively.The eTofts model shows a significant overestimation of pi in comparison with SSM in GBM(P<0.001),which is in accordance with MC simulations,and this overestimation shows dependence on the intra-to-extracellular water exchange rate constant(kio).Significant negative correlations between ADC and SSM-derived pi were found in both voxel-wise analyses(t-test P<0.001,average r=-0.74)and inter-subject comparisons(r=-0.63,P=0.009).But no consistent voxel-wise correlations(P>0.05)and a weaker inter-subject negative correlation(r=-0.56,P=0.02)were found between ADC and eTofts-derived pi.Further experimental and FD results revealed that kio made a limited contribution to ADC values in the physiological kio range in GBM,supporting ADC as a valid biomarker of cell density.These results suggest that the DCE-MRI pharmacokinetic shutter-speed model could significantly improve its accuracy in cell density estimation because of the considering transmembrane water exchange.
