Blowup for Hyperbolic Equations. Helge Kristian Jenssen and Carlo Sinestrari
|
|
- Frank Armstrong
- 6 years ago
- Views:
Transcription
1 Blowup for Hyperbolic Equations Helge Kristian Jenssen and Carlo Sinestrari Abstract. We consider dierent situations of blowup in sup-norm for hyperbolic equations. For scalar conservation laws with a source the asymptotic prole of the solution close to a blowup point is described in detail. Based on an example of Jerey we next show how blowup for ordinary dierential equations can be used to construct examples of blowup for systems of hyperbolic equations. Finally we outline the construction of solutions to certain strictly hyperbolic 3 3-systems of conservation laws which blow up in either sup-norm or total variation norm in nite time. 1. Scalar Equations 1.1. Introduction In this rst section we consider blowup in sup-norm for scalar equations. For a more complete discussion including proofs see [6]. We study blowup of positive solutions of the scalar conservation law with t u(x t)+@ x f(u(x t)) = g(u(x t)) u(x 0) = u 0 (x): (1) When the source term g(u) grows fast enough the solution may blow up at a nite point in the sense that there is a positive T<+1 and an X 2 (;1 1) such that lim sup u(x t) =+1 x!x t"t We refer to (X T) asablowup point. Our goal is to provide detailed information on how the solution behaves near such a point. We consider this as a rst step towards the understanding of blowup for physically relevant systems with sources. The present work is inspired by the similar analysis by Bressan [2] for a semilinear reactive{euler system. The problems of nding conditions under which the solution of (1) blows up, of determining whether the blowup is \localized", and to which extent the solution can be continued beyond the blowup time T have been studied by several authors, see [1], [10], and [11]. In the present work we bring the analysis one step further, i.e. we give a general procedure to study the asymptotic prole of solutions of (1) near a blowup point. For simplicity we assume that the initial value changes monotonicity only once, having a single maximum point with strictly negative
2 294 Helge Kristian Jenssen and Carlo Sinestrari second derivative. The analysis is complicated by the fact that solutions of (1) may develop discontinuities (shocks) in nite time. We consequently divide the analysis into dierent cases according to whether the solution blows up along a shock curve or in a region of smoothness. We refer to these cases as the shock case and smooth case, respectively. The main tool in our analysis is Dafermos' theory of generalized characteristics [3]. Our results can be summed up as follows. In the shock case the solution converges to an unbounded traveling wave on one side of the shock, while it remains bounded on the other side. In the smooth case the solution tends asymptotically to a function constant on parabolas about the blowup point. We also establish that the smooth case can occur only if the source term grows fast enough. As an application of our results we discuss two specic examples where the source term g is either exponential or powerlike Preliminaries We assume that the following conditions are satised for some k 3. (H1): f is strictly convex, f 0 (0)=0,andf 2 C 1 ([0 1)) \ C k+1 ((0 1)). (H2): g 2 C 1 ([0 1)) \ C k ((0 1)) and g(u) > 0 for any u>0. In addition Z 1 1 f 0 (v) dv < 1: (2) g(v) (H3): u 0 is nonnegative and of class C k, is increasing in (;1 x ] and decreasing in [x 1) for some x 2 R. (H4): For any x<x such that u 0 (x) > 0wehave u 0 0(x) > 0 moreover u 00 0(x ) < 0. These assumptions ensure the existence and uniqueness of an entropy solution of problem (1), at least locally in time. The solution is nonnegative and the solution has a maximal interval of existence [0 T), with 0 <T +1. We assume in the following that T<1. Let () denote the forward characteristic starting at (x 0). The following results form the basis for the detailed analysis of the blowup prole. Theorem 1.1. The curve (t) tends to a limit X<+1 as t " T.Moreover, there is a classical characteristic ending at (X T) and u! +1 along. Theorem 1.2. If the characteristic curve is a shock in some interval [T 0 T), then u is uniformly bounded in the region f(x t) : t<t x>(t)g. Theorem 1.3. If N is a neighbourhood of(x T), then u is uniformly bounded in (R [0 T))n N. Corollary 1.4. There is a unique way to dene u( T):R! [0 +1], such that u( T) is continuous from the left, u(x T) =+1, u(x T ) < +1 for x 6= X, and such that u( t)! u( T) in the L 1 loc (RnfXg)-norm as t " T.
3 Blowup for Hyperbolic Equations 295 Any set of the form U \ (R [0 T] nf(x T)g) where U is a neighbourhood of (X T) isreferredtoasabackward neighbourhood of (X T). We nowlistthe possible behaviours of the solution near the blowup point. (I): The characteristic isashock for t close enough to T, i.e. u((t); t) > u((t)+ t) for all t greater than some T 0 <T. (II): The solution u is smooth in some backward neighbourhood of the blowup point (X T) and lim u(x t) =+1: (3) x!x t"t (III): The characteristic is classical in [0 T) but either any neighbourhood of (X T) contains points where u is discontinuous, or u tends to dierent values along dierent characteristics ending at (X T). We next give some results and examples for the two rst cases. For further details and a discussion of case (III) we refer to [6] Shock Case Let us set u = u 0 (x). The problem (1) has a local C k (;" ") solution both forward and backward in time. Hence there exists ">0 and a function 2 C k (;" ") such that (0) = 0 and u(x + (t) t) u for any t 2 (;" "). Also let c = 0 (0) = f 0 (u) ; g(u)p ;1. Theorem 1.5. Assume that (H1){(H4) are satised and suppose that the solution u of problem (1) blows up at time T and that the forward characteristic x = (t) from the point (x 0) is a shock for t close to T. Then Z +1 f 0 (v) ; c dv = X ; x ; c(t ; t)+o(jx ; x ; c(t ; t)j 2 ) (4) g(v) u(x t) for all (x t) with x (t), while u(x t) is uniformly bounded forx>(t). Observe that, up to lower order terms, formula (4) denes u implicitly as a traveling wave with speed c. Example 1.6. Exponential case. Consider the equation (1) with f(u) =u m =m for some m>1 and g(u) =e u.inthiscase we have Z 1 Z 1 u f 0 (v) ; c g(v) dv = u v m;1 ; c e v dv = e ;u u m;1 (1 + o(1)): Using (4) and setting = X ; x ; c(t ; t) we nd the following expression for the asymptotic prole of u(x t) as (x t)! (X T) with x (t): u(x t) =; ln +(m ; 1) ln(; ln )+o(1): (5)
4 296 Helge Kristian Jenssen and Carlo Sinestrari 1.4. Smooth Case The possibility of having smooth blowup depends on the relative strengths of the source and convection terms. We rst give a denition. Denition 1.7. Assume that f and g satisfy assumptions (H1){(H2) and let p = lim u!1 f 0 (u). Wesaythatg has subcritical growth with respect to f if p =+1 and f lim 00 (v)g(v) v!+1 f 0 =0: (6) (v) 3 If the limit in (6) is +1 or if p < +1 we say that g has supercritical growth. One can show that if the source has subcritical growth, then the blowup is necessarily of type (I). However for supercritical sources one can have smooth blowup. We restrict ourselves to case (II). Lemma 1.8. If g is supercritical and the solution u of (1) has the behaviour (II), then the translates of the level curve of u through (x 0) along are again level curves of u and cover univalently N. Theorem 1.9. Let assumptions (H1){(H4) be satised and let g have supercritical growth. Suppose that the solution u of problem (1) has the behaviour (II). Then, as (x t)! (X T), Z +1 dv g(v) =(T ; t + 2 (X ; x)2 )(1 + o(1)) (7) u(x t) where = ;u 00 0(x )=g(u). Example Power case. Consider equation (1) with f(u) =u m =m and g(u) = u p (m>1). Using the theorem above one can show that the solution can have smooth blowup only if p 2m ; 1 in addition, if p>2m ; 1, then where w =(p ; 1) 1 p;1. u(x t)=w (T ; t)+ 2 (X ; x)2 1 1;p (1 + o(1)) 2. Systems of Equations 2.1. Introduction The above results show that one has very detailed information about the blowup pattern for scalar hyperbolic equations. Much less is known for systems of equations and the situation here is far from fully understood. Even simple examples give insight about the possible mechanisms of blowup. The phenomenon of resonance and magnication for systems of hyperbolic conservation laws has recently been studied by several authors [8, 9, 12]. Young [12] constructs exact solutions to 3 3-systems of conservation laws. Depending on the choice of initial data and the interaction coecients one can construct
5 Blowup for Hyperbolic Equations 297 dierent types of behaviour such as arbitrary large magnication of total variation and p-norms (1 p 1) in nite time. The eigenvalues are constant so that these systems are linearly degenerate in each family. In[8] Joly, Metivier, and Rauch consider examples of systems which are genuinely nonlinear in all three elds. Building on the work by Rosales and Majda [9] and using the theory of weakly nonlinear geometric optics [7] they give examples of solutions for which the variation grows arbitrarily large and the sup-norm is amplied by arbitrarily large factors in nite time. Common to these two works is the use of data with small sup-norm. The results are local in the sense that to have large amplication of the data one has to prescribe data with correspondingly small sup-norm. The aim of this section is to present two other types of blowup for systems. The rst is a generalization of an example given by Jerey [4] and illustrates how ordinary dierential equations can be used to construct blowup patterns for sysetms of hyperbolic equations. In the nal section we give a class of 3 3- systems of conservation laws with solutions whose sup-norm or total variation become innite in nite time Degenerate Hyperbolic Systems The object of this section is to understand the blowup mechanism for a particular class of systems and establish to which extent it is particular to 3 3-systems of hyperbolic equations. We also consider the possibility of writing the systems in conservative form. In [4] Jerey considered a quasilinear 3 3-system of the form U t + M(U)U x =0 (8) where U = U(x t) =(u(x t) v(x t) w(x t)) T, and M(U) is a strictly hyperbolic matrix (i.e. M(U) has three real and distinct eigenvalues) which depends nonlinearly on U. The system is in nonconservative form, the data is given on a compact interval, and the system is linearly degenerate in each eld (i.e. each eigenvalue is constant along the integral curves of the corresponding eigenvector elds). In Jerey's example the matrix M(U) is given by M(U) = ; cosh(2v) 0 ; sinh(2v) cosh(v) 0 sinh(v) sinh(2v) 0 cosh(2v) 1 A (9) where is a non-negative parameter. The eigenvalues of M(U) are 1 and 0. With initial data U(x 0) = (x=(h) 0 ;x=(h)) on the compact interval [;h h], an explicit solution U (x t) isgiven by u (x t) = ;1 1 1;t + x ; 1 v (x t) =lnj1 ; tj w (x t) = ;1 1 1;t ; x ; 1 :
6 298 Helge Kristian Jenssen and Carlo Sinestrari Here >0 is a constant and = =(h). As observed in [4], the vector U represents the solution only in the domain of determinacy D given by D = f(x t) :0 t h ;jxjg For the set D intersects the critical time-line t = t c = ;1 where the solution becomes unbounded. Thus, when, this provides an example of a strictly hyperbolic 3 3-system for which the sup-norm (and consequently also the total variation) tends to innity in nite time. To analyze the example given by Jerey, we note some key properties of the system given above. The matrix M(U) depends on v only and the entries along the middle column are identically zero. Also, the rst and third components of the solution are linear in x (with opposite coecients) while the middle component v of the solution is a function of t alone. Starting with these properties we see what kind of systems we can construct. Thus, suppose the rst and third components of the solution are of the form u(x t)=x +~u(t) w(x t) =;x +~w(t) where is a real constant. Observe that u(x 0) and w(x 0) are then necessarily linear ane functions of x. Thus if the initial data are to have nite total variation or sup-norm, then they must be prescribed on a compact interval. Next let the matrix M(U) have the form M(U) = The system (8) then takes the form a(v) 0 A(v) b(v) 0 B(v) c(v) 0 C(v) ~u 0 + (a(v) ; A(v)) = 0 v t + (b(v) ; B(v)) = 0 ~w 0 + (c(v) ; C(v))=0 1 A : (10) where prime denotes dierentiation with respect to t. If we in addition assume that v depends only on t, then this is a system of ordinary dierential equations where the second equation is decoupled from the other two. One can thus construct blowup for the system (12) simply by giving coecients, b(v), and B(v) such that the second equation blows up in nite time. However, we also want the matrix M(U) tobehyperbolic and that the blowup occurs within the domain of determinacy. A simple way of doing this is to prescribe constant eigenvalues (as in Jerey's example) and then adjusting the compact interval where the initial data are given so that the domain of determinacy intersects the critical timeline t = t c where the solution of the second equation blows up. Notice that 0 is always an eigenvalue of M. Also, M has eigenvalues 0 ( 0 being a positive constant) if
7 Blowup for Hyperbolic Equations 299 and only if the coecients a(v), A(v), c(v), and C(v) satisfy a(v) =;C(v) A(v)c(v) = 2 0 ; a(v) 2 : These relations are independent of the coecients in the second equation, which makes it easy to produce examples of the same type as that of Jerey. Example 2.1. Let 0 = =1, B(v) =;b(v) =v 2 =2, and assume that v(x 0) = 1 on the compact interval [;h h], where h 1. Inthiscase v satises the ordinary dierential equation Hence _v = v 2 v(0)=1: v(x t) =v(t) = 1 1 ; t which blows up at time t c =1. The choice for h guarantees that the domain of determinacy intersects the critical timeline t = t c. There are many ways to choose the remaining coecients. For example let T (v) =1; v 1 and dene a(v) =;C(v) = cos(t (v)) With B and b as above this gives the system with an explicit solution given by A(v) =c(v) = sin(t (v)): u t + cos(t (v))u x +sin(t (v))w x =0 v t ; v2 2 u x + v2 2 w x =0 w t +sin(t (v))u x ; cos(t (v))w x =0 u(x t) =x ; cos(t) ; sin(t) v(x t) = 1 1;t w(x t) =;x + cos(t) ; sin(t): We observe that in this case only the v-component blows up. Thus we see that it is easy to construct examples of blowup for a large class of degenerate hyperbolic systems. Furthermore the blowup mechanism for these systems is essentially that of an ordinary dierential equation with a superlinear righthand side. As a consequence the blowup does not depend on the fact that the above systems are 3 3-systems. Indeed, using the same technique as above one can easily give examples of 2 2-systems with similar behavior. This is in contrast to the results in [5, 8, 12] for conservation laws where it is essential that one consider systems of three or more equations, see below. Finally we observe that the systems above can in fact be written on conservation form. This follows since
8 300 Helge Kristian Jenssen and Carlo Sinestrari the matrix M depends only on a component of the solution which is independent of x. Hence [M(U)U] x =[M(U)] x U + M(U)U x = M(U)U x such that if U satises the quasilinear system (8), then U also satises the conservative system U t +[M(U)U] x =0: (11) Note however that U is not necessarily a solution of (11) for all times t<t c.the domain of determinacy for (11) will in general be dierent from that of (8). In particular the maximal and minimal characteristics emanating from the left and right endpoints where the initial data are given may meet before the blowup time t = t c Systems of Conservation Laws In this section we present a class of 3 3-systems of conservation laws U t + F (U) x =0 (12) for which one can prescribe initial data such that the solution blows up in nite time. The main result is the following theorem. Theorem 2.2. There exist 33-systems of strictly hyperbolic conservation laws for which the solution U(x t) has one of the following properties. (a) There exists a time T, 0 <T<1, such that limku( t)k 1 =+1: (13) t"t (b) There exist a constant C>0 andatimet, 0 <T<1, such that limt.v. [U( t)]=+1 while ku( t)k 1 <C for all t<t: (14) t"t Outline of Construction The class of systems is a modication of the examples considered by Young [12]. We want to set up an interaction pattern where two 2-shocks approach each other while 1- and 3-waves (which will be contact discontinuities) are reected back and forth between the 2-shocks. Notethatthis requires at least three equations, i.e. it is not possible to get an interaction pattern like this for 2 2-systems. We dothisby constructing solutions to 3 3-systems of the form (12) where the ux function F has the form F (U) = ua(v)+w ;(v) u( 2 0 ; a 2 (v)) ; wa(v) 1 A : (15) Here 0 > 0 is a constant and a(v) will be chosen to obtain the various behaviour stated in Theorem 2.2. To simplify the analysis we assume that ;(v) has the following properties, (i) ;(v) is strictly convex (ii) ; 0 <(v) =; 0 (v) < 0 for all v 2 R
9 Blowup for Hyperbolic Equations 301 (iii) ;(0) = 0 and ;(;v) =;(v) for all v 2 R. It is readily checked that the eigenvalues of the Jacobian DF are 1 = ; 0 2 = (v) 3 =+ 0 : (16) Thus (ii) guarantees that the system is strictly hyperbolic. Note that the second equation in the system is a decoupled scalar conservation law for v with a strictly convex ux. It follows that the second characteristic eld is genuinely nonlinear. The rst and third elds are linearly degenerate so that all 1- and 3-waves are contact discontinuities. It follows that shock and rarefaction curves coincide in the rst and the third families, and these are straight lines in planes with v = constant. The interaction pattern is constructed by prescribing initial data with four constant states l, m, M, and r (ordered from left to right). The states are chosen so that the Riemann problems (l m), (m M), and (M r) are solved by a single 2- shock, a single 1-wave, and a single 2-shock, respectively. At some later time the left 2-shock interacts with the 1-wave producing a transmitted 1-wave, a transmitted 2- shock, and a reected 3-wave. In turn this reected 3-wave interacts with the right 2-shock and gives a reected 1-wave, a transmitted 2-shock, and a transmitted 3-wave. The reected 1-wave then interacts with the left 2-shock, and so on. If we put v l = V > 0, v m = v M =0,andv r = ;V, then it follows by convexity of; that the 2-shocks will approach each other and meet in nite time t = T. Since each interaction yields a reected wave it follows that there are an innite number of interactions in nite time. The proof of the theorem is completed by suitably choosing V and a(v). To obtain the behavior described in part (a) of the theorem we let a(v) bealinear function. One can then show that the strength of the reected and transmitted waves in each interaction are larger than the strength of the incoming wave provided V is large enough. Since there is an innite number of interactions in nite time one thus get blowup in sup-norm by choosing V suciently large. To have the behavior described in part (b) we leta(v) be a quadratic function. For a suitable choice of V one can prove that the corresponding solution is periodic in state space. Again, since there is an innite number of interactions in nite time one conclude that while the sup-norm remains uniformly bounded the total variation of the solution tends to innity ast " T.We refer to [5] for the details of the computations. Acknowledgments. We thank A. Bressan for suggesting the problem of blowup for hyperbolic equations and for his kind interest. The rst author is indebted to A. Bressan, B. Piccoli, H. Holden, and N. H. Risebro for discussion on blowup for systems. References [1] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Birkhauser, Boston, 1995.
10 302 Helge Kristian Jenssen and Carlo Sinestrari [2] A. Bressan, Blowup Asymptotics for the Reactive-Euler Gas Model, SIAM J. Math. Anal., 22 (1992), 587{601. [3] C. M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J., 26 (1977), 1097{1119. [4] A. Jeffrey, Breakdown of the Solution to a Completely Exceptional System of Hyperbolic Equations, J. Math. Anal. Appl., 45 (1974), 375{381. [5] H. K. Jenssen, Blowup for Systems of Conservation Laws, preprint 1998, available at [6] H. K. Jenssen, C. Sinestrari Blowup asymptotics for scalar conservation laws with a source, submitted, available at [7] J. L. Joly, G. Metivier, J. Rauch, Resonant one-dimensional nonlineargeometric optics, J. Funct. Anal., 114 (1993), no. 1, 106{231. [8] J. L. Joly, G. Metivier, J. Rauch, A nonlinear instability for 3 3 systems of conservation laws, Comm. Math. Phys., 162 (1994), no. 1, 47{59. [9] A. Majda, R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves. I. A single space variable, Stud. Appl. Math., 71 (1984), no. 2, 149{179. [10] R. Natalini, C. Sinestrari, A. Tesei, Incomplete blow-up of solutions of quasilinear balance laws, Arch. Rat. Mech. Anal., 135 (1996), 259{296. [11] R. Natalini, A. Tesei, Blow{up of solutions for a class of balance laws, Comm. Partial Dierential Equations, 19 (1994), 417{453. [12] R. Young, Exact Solutions to Degenerate Conservation Laws, preprint Department of Mathematical Sciences, Norwegian University of Science and Technology, NTNU, N-7034 Trondheim, Norway address: helgekj@math.ntnu.no Dipartimento di Matematica, Universita di Roma \Tor Vergata", Via della Ricerca Scientica, Roma, Italy. address: sinestra@axp.mat.utovrm.it
Applications of the compensated compactness method on hyperbolic conservation systems
Applications of the compensated compactness method on hyperbolic conservation systems Yunguang Lu Department of Mathematics National University of Colombia e-mail:ylu@unal.edu.co ALAMMI 2009 In this talk,
More informationPointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang
Pointwise convergence rate for nonlinear conservation laws Eitan Tadmor and Tao Tang Abstract. We introduce a new method to obtain pointwise error estimates for vanishing viscosity and nite dierence approximations
More informationInstability of Finite Difference Schemes for Hyperbolic Conservation Laws
Instability of Finite Difference Schemes for Hyperbolic Conservation Laws Alberto Bressan ( ), Paolo Baiti ( ) and Helge Kristian Jenssen ( ) ( ) Department of Mathematics, Penn State University, University
More informationOn the Front-Tracking Algorithm
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 7, 395404 998 ARTICLE NO. AY97575 On the Front-Tracking Algorithm Paolo Baiti S.I.S.S.A., Via Beirut 4, Trieste 3404, Italy and Helge Kristian Jenssen
More informationAbstract. A front tracking method is used to construct weak solutions to
A Front Tracking Method for Conservation Laws with Boundary Conditions K. Hvistendahl Karlsen, K.{A. Lie, and N. H. Risebro Abstract. A front tracking method is used to construct weak solutions to scalar
More informationON SCALAR CONSERVATION LAWS WITH POINT SOURCE AND STEFAN DIEHL
ON SCALAR CONSERVATION LAWS WITH POINT SOURCE AND DISCONTINUOUS FLUX FUNCTION STEFAN DIEHL Abstract. The conservation law studied is @u(x;t) + @ F u(x; t); x @t @x = s(t)(x), where u is a f(u); x > 0 concentration,
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More information2 The second case, in which Problem (P 1 ) reduces to the \one-phase" problem (P 2 ) 8 >< >: u t = u xx + uu x t > 0, x < (t) ; u((t); t) = q t > 0 ;
1 ON A FREE BOUNDARY PROBLEM ARISING IN DETONATION THEORY: CONVERGENCE TO TRAVELLING WAVES 1. INTRODUCTION. by M.Bertsch Dipartimento di Matematica Universita di Torino Via Principe Amedeo 8 10123 Torino,
More informationSingularity formation for compressible Euler equations
Singularity formation for compressible Euler equations Geng Chen Ronghua Pan Shengguo Zhu Abstract In this paper, for the p-system and full compressible Euler equations in one space dimension, we provide
More informationMODELLING OF FLEXIBLE MECHANICAL SYSTEMS THROUGH APPROXIMATED EIGENFUNCTIONS L. Menini A. Tornambe L. Zaccarian Dip. Informatica, Sistemi e Produzione
MODELLING OF FLEXIBLE MECHANICAL SYSTEMS THROUGH APPROXIMATED EIGENFUNCTIONS L. Menini A. Tornambe L. Zaccarian Dip. Informatica, Sistemi e Produzione, Univ. di Roma Tor Vergata, via di Tor Vergata 11,
More informationON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay
ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov
More informationOn the Dependence of Euler Equations on Physical Parameters
On the Dependence of Euler Equations on Physical Parameters Cleopatra Christoforou Department of Mathematics, University of Houston Joint Work with: Gui-Qiang Chen, Northwestern University Yongqian Zhang,
More informationOn the Cauchy Problems for Polymer Flooding with Gravitation
On the Cauchy Problems for Polymer Flooding with Gravitation Wen Shen Mathematics Department, Penn State University. Email: wxs27@psu.edu November 5, 2015 Abstract We study two systems of conservation
More informationII. Systems of viscous hyperbolic balance laws. Bernold Fiedler, Stefan Liebscher. Freie Universitat Berlin. Arnimallee 2-6, Berlin, Germany
Generic Hopf bifurcation from lines of equilibria without parameters: II. Systems of viscous hyperbolic balance laws Bernold Fiedler, Stefan Liebscher Institut fur Mathematik I Freie Universitat Berlin
More informationPresenter: Noriyoshi Fukaya
Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), 347-373. Presenter: Noriyoshi
More informationL 1 stability of conservation laws for a traffic flow model
Electronic Journal of Differential Equations, Vol. 2001(2001), No. 14, pp. 1 18. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
More informationTHE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: REGULARITY AND LONG TIME BEHAVIOR OF WEAK SOLUTIONS A.L. BERTOZZI AND M. PUGH.
THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: REGULARITY AND LONG TIME BEHAVIOR OF WEAK SOLUTIONS A.L. BERTOI AND M. PUGH April 1994 Abstract. We consider the fourth order degenerate diusion equation
More informationGlobal unbounded solutions of the Fujita equation in the intermediate range
Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,
More informationNonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationTravelling waves. Chapter 8. 1 Introduction
Chapter 8 Travelling waves 1 Introduction One of the cornerstones in the study of both linear and nonlinear PDEs is the wave propagation. A wave is a recognizable signal which is transferred from one part
More informationREGULARITY THROUGH APPROXIMATION FOR SCALAR CONSERVATION LAWS
SIAM J. MATH. ANAL. c 1988 Society for Industrial and Applied Mathematics Vol. 19, No. 4, pp. 1 XX, July 1988 003 REGULARITY THROUGH APPROXIMATION FOR SCALAR CONSERVATION LAWS BRADLEY J. LUCIER Abstract.
More informationGeneric Singularities of Solutions to some Nonlinear Wave Equations
Generic Singularities of Solutions to some Nonlinear Wave Equations Alberto Bressan Deartment of Mathematics, Penn State University (Oberwolfach, June 2016) Alberto Bressan (Penn State) generic singularities
More informationShock Waves in Plane Symmetric Spacetimes
Communications in Partial Differential Equations, 33: 2020 2039, 2008 Copyright Taylor & Francis Group, LLC ISSN 0360-5302 print/1532-4133 online DOI: 10.1080/03605300802421948 Shock Waves in Plane Symmetric
More informationUNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9947(XX)0000-0 UNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS YULIAN
More informationSome nonlinear elliptic equations in R N
Nonlinear Analysis 39 000) 837 860 www.elsevier.nl/locate/na Some nonlinear elliptic equations in Monica Musso, Donato Passaseo Dipartimento di Matematica, Universita di Pisa, Via Buonarroti,, 5617 Pisa,
More informationHyperbolic Systems of Conservation Laws. in One Space Dimension. II - Solutions to the Cauchy problem. Alberto Bressan
Hyperbolic Systems of Conservation Laws in One Space Dimension II - Solutions to the Cauchy problem Alberto Bressan Department of Mathematics, Penn State University http://www.math.psu.edu/bressan/ 1 Global
More informationLectures 15: Parallel Transport. Table of contents
Lectures 15: Parallel Transport Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this lecture we study the
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationGerhard Rein. Mathematisches Institut der Universitat Munchen, Alan D. Rendall. Max-Planck-Institut fur Astrophysik, and
A regularity theorem for solutions of the spherically symmetric Vlasov-Einstein system Gerhard Rein Mathematisches Institut der Universitat Munchen, Theresienstr. 39, 80333 Munchen, Germany, Alan D. Rendall
More informationContractive metrics for scalar conservation laws
Contractive metrics for scalar conservation laws François Bolley 1, Yann Brenier 2, Grégoire Loeper 34 Abstract We consider nondecreasing entropy solutions to 1-d scalar conservation laws and show that
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationOn the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist
Nonlinear Analysis 49 (2002) 603 611 www.elsevier.com/locate/na On the role playedby the Fuck spectrum in the determination of critical groups in elliptic problems where the asymptotic limits may not exist
More informationExistence Theory for Hyperbolic Systems of Conservation Laws with General Flux-Functions
Existence Theory for Hyperbolic Systems of Conservation Laws with General Flux-Functions Tatsuo Iguchi & Philippe G. LeFloch Abstract For the Cauchy problem associated with a nonlinear, strictly hyperbolic
More informationBlow-up for a Nonlocal Nonlinear Diffusion Equation with Source
Revista Colombiana de Matemáticas Volumen 46(2121, páginas 1-13 Blow-up for a Nonlocal Nonlinear Diffusion Equation with Source Explosión para una ecuación no lineal de difusión no local con fuente Mauricio
More information2 SCHECTER is a step function, then in the sense of distributions, f(u) x is the measure (f(u + ) f(u ))(x): (1.6) This leads to the Rankine-Hugoniot
TRAVELING-WAVE SOLUTIONS OF CONVECTION-DIFFUSION SYSTEMS BY CENTER MANIFOLD REDUCTION STEPHEN SCHECTER Abstract. Traveling waves u(x st) for systems of conservation laws u t + Df(u)u x = (B(u)u x ) x were
More informationMultiplicity ofnontrivial solutions for an asymptotically linear nonlocal Tricomi problem
Nonlinear Analysis 46 (001) 591 600 www.elsevier.com/locate/na Multiplicity ofnontrivial solutions for an asymptotically linear nonlocal Tricomi problem Daniela Lupo 1, Kevin R. Payne ; ; Dipartimento
More informationarxiv: v2 [math.ap] 1 Jul 2011
A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime arxiv:1105.3074v2 [math.ap] 1 Jul 2011 Abstract Philippe G. efloch 1 and Mai Duc Thanh 2 1 aboratoire
More informationM. HERTY, CH. JÖRRES, AND B. PICCOLI
EXISTENCE OF SOLUTION TO SUPPLY CHAIN MODELS BASED ON PARTIAL DIFFERENTIAL EQUATION WITH DISCONTINUOUS FLUX FUNCTION M. HERTY, CH. JÖRRES, AND B. PICCOLI Abstract. We consider a recently [2] proposed model
More informationRearrangements and polar factorisation of countably degenerate functions G.R. Burton, School of Mathematical Sciences, University of Bath, Claverton D
Rearrangements and polar factorisation of countably degenerate functions G.R. Burton, School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K. R.J. Douglas, Isaac Newton
More informationDepartment of Mathematics. University of Notre Dame. Abstract. In this paper we study the following reaction-diusion equation u t =u+
Semilinear Parabolic Equations with Prescribed Energy by Bei Hu and Hong-Ming Yin Department of Mathematics University of Notre Dame Notre Dame, IN 46556, USA. Abstract In this paper we study the following
More informationt x 0.25
Journal of ELECTRICAL ENGINEERING, VOL. 52, NO. /s, 2, 48{52 COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS Ivan Cimrak If the time discretization of a nonlinear parabolic
More informationCongurations of periodic orbits for equations with delayed positive feedback
Congurations of periodic orbits for equations with delayed positive feedback Dedicated to Professor Tibor Krisztin on the occasion of his 60th birthday Gabriella Vas 1 MTA-SZTE Analysis and Stochastics
More informationLectures 18: Gauss's Remarkable Theorem II. Table of contents
Math 348 Fall 27 Lectures 8: Gauss's Remarkable Theorem II Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams.
More informationCentral Schemes for Systems of Balance Laws Salvatore Fabio Liotta, Vittorio Romano, Giovanni Russo Abstract. Several models in mathematical physics a
Central Schemes for Systems of Balance Laws Salvatore Fabio Liotta, Vittorio Romano, Giovanni Russo Abstract. Several models in mathematical physics are described by quasilinear hyperbolic systems with
More informationRadon measure solutions for scalar. conservation laws. A. Terracina. A.Terracina La Sapienza, Università di Roma 06/09/2017
Radon measure A.Terracina La Sapienza, Università di Roma 06/09/2017 Collaboration Michiel Bertsch Flavia Smarrazzo Alberto Tesei Introduction Consider the following Cauchy problem { ut + ϕ(u) x = 0 in
More informationOn non negative solutions of some quasilinear elliptic inequalities
On non negative solutions of some quasilinear elliptic inequalities Lorenzo D Ambrosio and Enzo Mitidieri September 28 2006 Abstract Let f : R R be a continuous function. We prove that under some additional
More informationUniqueness of ground states for quasilinear elliptic equations in the exponential case
Uniqueness of ground states for quasilinear elliptic equations in the exponential case Patrizia Pucci & James Serrin We consider ground states of the quasilinear equation (.) div(a( Du )Du) + f(u) = 0
More informationThis method is introduced by the author in [4] in the case of the single obstacle problem (zero-obstacle). In that case it is enough to consider the v
Remarks on W 2;p -Solutions of Bilateral Obstacle Problems Srdjan Stojanovic Department of Mathematical Sciences University of Cincinnati Cincinnati, OH 4522-0025 November 30, 995 Abstract The well known
More informationQUADRATIC RATE OF CONVERGENCE FOR CURVATURE DEPENDENT SMOOTH INTERFACES: A SIMPLE PROOF 1
QUADRATIC RATE OF CONVERGENCE FOR CURVATURE DEPENDENT SMOOTH INTERFACES: A SIMPLE PROOF R. H. NOCHETTO Department of Mathematics and Institute for Physical Science and Technology University of Maryland,
More informationBifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces
Nonlinear Analysis 42 (2000) 561 572 www.elsevier.nl/locate/na Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces Pavel Drabek a;, Nikos M. Stavrakakis b a Department
More informationInvariant Sets for non Classical Reaction-Diffusion Systems
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 1, Number 6 016, pp. 5105 5117 Research India Publications http://www.ripublication.com/gjpam.htm Invariant Sets for non Classical
More informationIterative procedure for multidimesional Euler equations Abstracts A numerical iterative scheme is suggested to solve the Euler equations in two and th
Iterative procedure for multidimensional Euler equations W. Dreyer, M. Kunik, K. Sabelfeld, N. Simonov, and K. Wilmanski Weierstra Institute for Applied Analysis and Stochastics Mohrenstra e 39, 07 Berlin,
More informationA Very Brief Introduction to Conservation Laws
A Very Brief Introduction to Wen Shen Department of Mathematics, Penn State University Summer REU Tutorial, May 2013 Summer REU Tutorial, May 2013 1 / The derivation of conservation laws A conservation
More informationSPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS
SPECTRAL PROPERTIES AND NODAL SOLUTIONS FOR SECOND-ORDER, m-point, BOUNDARY VALUE PROBLEMS BRYAN P. RYNNE Abstract. We consider the m-point boundary value problem consisting of the equation u = f(u), on
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Delta and Singular Delta Locus for One Dimensional Systems of Conservation Laws M. Nedeljkov
More information2 BAISHENG YAN When L =,it is easily seen that the set K = coincides with the set of conformal matrices, that is, K = = fr j 0 R 2 SO(n)g: Weakly L-qu
RELAXATION AND ATTAINMENT RESULTS FOR AN INTEGRAL FUNCTIONAL WITH UNBOUNDED ENERGY-WELL BAISHENG YAN Abstract. Consider functional I(u) = R jjdujn ; L det Duj dx whose energy-well consists of matrices
More informationSimple waves and characteristic decompositions of quasilinear hyperbolic systems in two independent variables
s and characteristic decompositions of quasilinear hyperbolic systems in two independent variables Wancheng Sheng Department of Mathematics, Shanghai University (Joint with Yanbo Hu) Joint Workshop on
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More information1 Introduction We will consider traveling waves for reaction-diusion equations (R-D) u t = nx i;j=1 (a ij (x)u xi ) xj + f(u) uj t=0 = u 0 (x) (1.1) w
Reaction-Diusion Fronts in Periodically Layered Media George Papanicolaou and Xue Xin Courant Institute of Mathematical Sciences 251 Mercer Street, New York, N.Y. 10012 Abstract We compute the eective
More informationThe Best Circulant Preconditioners for Hermitian Toeplitz Systems II: The Multiple-Zero Case Raymond H. Chan Michael K. Ng y Andy M. Yip z Abstract In
The Best Circulant Preconditioners for Hermitian Toeplitz Systems II: The Multiple-ero Case Raymond H. Chan Michael K. Ng y Andy M. Yip z Abstract In [0, 4], circulant-type preconditioners have been proposed
More informationOn the Cauchy Problems for Polymer Flooding with Gravitation
On the Cauchy Problems for Polymer Flooding with Gravitation Wen Shen Mathematics Department, Penn State University Abstract We study two systems of conservation laws for polymer flooding in secondary
More informationThe inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method
The inviscid limit to a contact discontinuity for the compressible Navier-Stokes-Fourier system using the relative entropy method Alexis Vasseur, and Yi Wang Department of Mathematics, University of Texas
More informationFractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization
Progress in Nonlinear Differential Equations and Their Applications, Vol. 63, 217 224 c 2005 Birkhäuser Verlag Basel/Switzerland Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization
More informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationmaximally charged black holes and Hideki Ishihara Department ofphysics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152, Japan
Quasinormal modes of maximally charged black holes Hisashi Onozawa y,takashi Mishima z,takashi Okamura, and Hideki Ishihara Department ofphysics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo
More informationFLUX IDENTIFICATION IN CONSERVATION LAWS 2 It is quite natural to formulate this problem more or less like an optimal control problem: for any functio
CONVERGENCE RESULTS FOR THE FLUX IDENTIFICATION IN A SCALAR CONSERVATION LAW FRANCOIS JAMES y AND MAURICIO SEP ULVEDA z Abstract. Here we study an inverse problem for a quasilinear hyperbolic equation.
More informationShock formation in the compressible Euler equations and related systems
Shock formation in the compressible Euler equations and related systems Geng Chen Robin Young Qingtian Zhang Abstract We prove shock formation results for the compressible Euler equations and related systems
More informationLinear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space
Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................
More information4.1 Eigenvalues, Eigenvectors, and The Characteristic Polynomial
Linear Algebra (part 4): Eigenvalues, Diagonalization, and the Jordan Form (by Evan Dummit, 27, v ) Contents 4 Eigenvalues, Diagonalization, and the Jordan Canonical Form 4 Eigenvalues, Eigenvectors, and
More informationHyperbolic Gradient Flow: Evolution of Graphs in R n+1
Hyperbolic Gradient Flow: Evolution of Graphs in R n+1 De-Xing Kong and Kefeng Liu Dedicated to Professor Yi-Bing Shen on the occasion of his 70th birthday Abstract In this paper we introduce a new geometric
More informationu-= (u, v), x>o, j u, (u,, v,), x<o, U(X 0) (1) (1), A A2 only when u 0, in which case A 0. THE RIEMANN PROBLEM NEAR A HYPERBOLIC SINGULARITY II*
SIAM J. APPL. MATH. Vol. 48, No. 6, December 1988 (C) 1988 Society for Industrial and Applied Mathematics 006 THE RIEMANN PROBLEM NEAR A HYPERBOLIC SINGULARITY II* E. ISAACSONS" AN[) B. TEMPLE:I: Abstract.
More informationWEAK ASYMPTOTIC SOLUTION FOR A NON-STRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS-II
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 94, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu WEAK ASYMPTOTIC
More informationHyperbolic Systems of Conservation Laws
Hyperbolic Systems of Conservation Laws III - Uniqueness and continuous dependence and viscous approximations Alberto Bressan Mathematics Department, Penn State University http://www.math.psu.edu/bressan/
More informationSimple waves and a characteristic decomposition of the two dimensional compressible Euler equations
Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations Jiequan Li 1 Department of Mathematics, Capital Normal University, Beijing, 100037 Tong Zhang Institute
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-19 Wien, Austria The Negative Discrete Spectrum of a Class of Two{Dimentional Schrodinger Operators with Magnetic
More informationLocalization phenomena in degenerate logistic equation
Localization phenomena in degenerate logistic equation José M. Arrieta 1, Rosa Pardo 1, Anibal Rodríguez-Bernal 1,2 rpardo@mat.ucm.es 1 Universidad Complutense de Madrid, Madrid, Spain 2 Instituto de Ciencias
More informationMath Partial Differential Equations 1
Math 9 - Partial Differential Equations Homework 5 and Answers. The one-dimensional shallow water equations are h t + (hv) x, v t + ( v + h) x, or equivalently for classical solutions, h t + (hv) x, (hv)
More informationConstrained Leja points and the numerical solution of the constrained energy problem
Journal of Computational and Applied Mathematics 131 (2001) 427 444 www.elsevier.nl/locate/cam Constrained Leja points and the numerical solution of the constrained energy problem Dan I. Coroian, Peter
More informationComments on integral variants of ISS 1
Systems & Control Letters 34 (1998) 93 1 Comments on integral variants of ISS 1 Eduardo D. Sontag Department of Mathematics, Rutgers University, Piscataway, NJ 8854-819, USA Received 2 June 1997; received
More informationL 2 Geometry of the Symplectomorphism Group
University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence
More informationAN OVERVIEW ON SOME RESULTS CONCERNING THE TRANSPORT EQUATION AND ITS APPLICATIONS TO CONSERVATION LAWS
AN OVERVIEW ON SOME RESULTS CONCERNING THE TRANSPORT EQUATION AND ITS APPLICATIONS TO CONSERVATION LAWS GIANLUCA CRIPPA AND LAURA V. SPINOLO Abstract. We provide an informal overview on the theory of transport
More informationSpurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics
UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Spurious Chaotic Solutions of Dierential Equations Sigitas Keras DAMTP 994/NA6 September 994 Department of Applied Mathematics and Theoretical Physics
More informationThe Pacic Institute for the Mathematical Sciences http://www.pims.math.ca pims@pims.math.ca Surprise Maximization D. Borwein Department of Mathematics University of Western Ontario London, Ontario, Canada
More informationPropagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R
Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R P. Poláčik School of Mathematics, University of Minnesota Minneapolis, MN 55455 Abstract We consider semilinear
More informationGlobal Existence of Large BV Solutions in a Model of Granular Flow
This article was downloaded by: [Pennsylvania State University] On: 08 February 2012, At: 09:55 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered
More informationNew concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space
Lesson 6: Linear independence, matrix column space and null space New concepts: Span of a vector set, matrix column space (range) Linearly dependent set of vectors Matrix null space Two linear systems:
More informationAuthor(s) Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 41(1)
Title On the stability of contact Navier-Stokes equations with discont free b Authors Huang, Feimin; Matsumura, Akitaka; Citation Osaka Journal of Mathematics. 4 Issue 4-3 Date Text Version publisher URL
More information2 J. BRACHO, L. MONTEJANO, AND D. OLIVEROS Observe as an example, that the circle yields a Zindler carrousel with n chairs, because we can inscribe in
A CLASSIFICATION THEOREM FOR ZINDLER CARROUSELS J. BRACHO, L. MONTEJANO, AND D. OLIVEROS Abstract. The purpose of this paper is to give a complete classication of Zindler Carrousels with ve chairs. This
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationGLOBAL EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC SYSTEM
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 35, Number 4, April 7, Pages 7 7 S -99396)8773-9 Article electronically published on September 8, 6 GLOBAL EXISTENCE OF SOLUTIONS TO A HYPERBOLIC-PARABOLIC
More informationSPLITTING OF SEPARATRICES FOR (FAST) QUASIPERIODIC FORCING. splitting, which now seems to be the main cause of the stochastic behavior in
SPLITTING OF SEPARATRICES FOR (FAST) QUASIPERIODIC FORCING A. DELSHAMS, V. GELFREICH, A. JORBA AND T.M. SEARA At the end of the last century, H. Poincare [7] discovered the phenomenon of separatrices splitting,
More informationsystem of equations. In particular, we give a complete characterization of the Q-superlinear
INEXACT NEWTON METHODS FOR SEMISMOOTH EQUATIONS WITH APPLICATIONS TO VARIATIONAL INEQUALITY PROBLEMS Francisco Facchinei 1, Andreas Fischer 2 and Christian Kanzow 3 1 Dipartimento di Informatica e Sistemistica
More informationPeriodic solutions of weakly coupled superlinear systems
Periodic solutions of weakly coupled superlinear systems Alessandro Fonda and Andrea Sfecci Abstract By the use of a higher dimensional version of the Poincaré Birkhoff theorem, we are able to generalize
More informationA general theory of discrete ltering. for LES in complex geometry. By Oleg V. Vasilyev AND Thomas S. Lund
Center for Turbulence Research Annual Research Briefs 997 67 A general theory of discrete ltering for ES in complex geometry By Oleg V. Vasilyev AND Thomas S. und. Motivation and objectives In large eddy
More informationAsymptotic behavior of infinity harmonic functions near an isolated singularity
Asymptotic behavior of infinity harmonic functions near an isolated singularity Ovidiu Savin, Changyou Wang, Yifeng Yu Abstract In this paper, we prove if n 2 x 0 is an isolated singularity of a nonegative
More informationAnumerical and analytical study of the free convection thermal boundary layer or wall jet at
REVERSED FLOW CALCULATIONS OF HIGH PRANDTL NUMBER THERMAL BOUNDARY LAYER SEPARATION 1 J. T. Ratnanather and P. G. Daniels Department of Mathematics City University London, England, EC1V HB ABSTRACT Anumerical
More informationMath 660-Lecture 23: Gudonov s method and some theories for FVM schemes
Math 660-Lecture 3: Gudonov s method and some theories for FVM schemes 1 The idea of FVM (You can refer to Chapter 4 in the book Finite volume methods for hyperbolic problems ) Consider the box [x 1/,
More informationMathematical Institute, University of Utrecht. The problem of estimating the mean of an observed Gaussian innite-dimensional vector
On Minimax Filtering over Ellipsoids Eduard N. Belitser and Boris Y. Levit Mathematical Institute, University of Utrecht Budapestlaan 6, 3584 CD Utrecht, The Netherlands The problem of estimating the mean
More informationFor later use we dene a constraint-set Z k utilizing an expression for the state predictor Z k (x k )=fz k =( k k ): k 2 ^ k 2 ^ k = F k ( k x k )g 8
IDENTIFICATION AND CONVEITY IN OPTIMIZING CONTROL Bjarne A. Foss Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-734 Trondheim, Norway. Email: baf@itk.unit.no Tor
More information