@article{jenssen_kogan_2020, title={A mixed boundary value problem for u(xy) = f (x , y, u, u(x), u(y))}, volume={268}, ISSN={["1090-2732"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85076254291&partnerID=MN8TOARS}, DOI={10.1016/j.jde.2019.11.063}, abstractNote={Consider a single hyperbolic PDE uxy=f(x,y,u,ux,uy), with locally prescribed data: u along a non-characteristic curve M and ux along a non-characteristic curve N. We assume that M and N are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the (x,y)-plane. It is known that if M is located above N, then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when M lies below N, the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function f). The construction, via Picard iteration, makes use of a careful choice of additional u-data which are updated in each iteration step.}, number={12}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Jenssen, Helge Kristian and Kogan, Irina A.}, year={2020}, month={Jun}, pages={7535–7560} }
 @article{benfield_jenssen_kogan_2019, title={Jacobians with prescribed eigenvectors}, volume={65}, ISBN={1872-6984}, ISSN={0926-2245}, url={http://dx.doi.org/10.1016/j.difgeo.2019.03.008}, DOI={10.1016/j.difgeo.2019.03.008}, abstractNote={Let Ω⊂Rn be open and let R be a partial frame on Ω; that is, a set of m linearly independent vector fields prescribed on Ω (m≤n). We consider the issue of describing the set of all maps F:Ω→Rn with the property that each of the given vector fields is an eigenvector of the Jacobian matrix of F. By introducing a coordinate independent definition of the Jacobian, we obtain an intrinsic formulation of the problem, which leads to an overdetermined PDE system, whose compatibility conditions can be expressed in an intrinsic, coordinate independent manner. To analyze this system we formulate and prove a generalization of the classical Frobenius integrability theorems. The size and structure of the solution set of this system depends on the properties of the partial frame; in particular, whether or not it is in involution. A particularly nice subclass of involutive partial frames, called rich frames, can be completely analyzed. The involutive, non-rich case is somewhat harder to handle. We provide a complete answer in the case of m=3 and arbitrary n, as well as some general results for arbitrary m. The non-involutive case is far more challenging, and we only obtain a comprehensive analysis in the case n=3, m=2. Finally, we provide explicit examples illustrating the various possibilities.}, journal={Differential Geometry and its Applications}, publisher={Elsevier BV}, author={Benfield, Michael and Jenssen, Helge Kristian and Kogan, Irina A.}, year={2019}, month={Aug}, pages={108–146} }
 @article{bressan_jenssen_baiti_2006, title={An instability of the Godunov scheme}, volume={59}, ISSN={["0010-3640"]}, DOI={10.1002/cpa.20141}, abstractNote={We construct a solution to a 2 × 2 strictly hyperbolic system of conservation laws, showing that the Godunov scheme [13] can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing‐viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or L1‐stability estimates can in general be valid for finite difference schemes. © 2006 Wiley Periodicals, Inc.}, number={11}, journal={COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS}, author={Bressan, Alberto and Jenssen, Helge Kristian and Baiti, Paolo}, year={2006}, month={Nov}, pages={1604–1638} }
 @article{jenssen_lyng_williams_2005, title={Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion}, volume={54}, ISSN={["1943-5258"]}, DOI={10.1512/iumj.2005.54.2685}, abstractNote={We use the classical normal mode approach of hydrodynamic stability theory to define stability determinants (Evans functions) for multidimensional strong detonations in three commonly studied models of combustion: the full reactive Navier-Stokes (RNS) model, and the simpler Zeldovich-von Neumann-Doring (ZND) and Chapman-Jouguet (CJ) models. The determinants are functions of frequencies (‚;·), where ‚ is a complex variable dual to the time variable, and · 2 R di1 is dual to the transverse spatial variables. The zeros of these determinants in 0 correspond to perturbations that grow exponentially with time. The CJ determinant, ¢CJ(‚;·), turns out to be explicitly computable. The RNS and ZND determinants are impossible to compute explicitly, but we are able to compute their first-order low-frequency expansions with an error term that is uniformly small with re- spect to all possible (‚;·) directions. Somewhat surprisingly, this computation yields an Equivalence Theorem: the leading coecient in the expansions of both the RNS and ZND determinants is a constant multiple of ¢CJ! In this sense the low-frequency stability condi- tions for strong detonations in all three models are equivalent. By computing ¢CJ we are able to give low-frequency stability criteria valid for all three models in terms of the physical quantities: Mach number, Gruneisen coecient, compression ratio, and heat release. The Equivalence Theorem and its surrounding analysis is a step toward the rigorous theoretical justification of the CJ and ZND models as approximations to the full RNS model.}, number={1}, journal={INDIANA UNIVERSITY MATHEMATICS JOURNAL}, author={Jenssen, HK and Lyng, G and Williams, M}, year={2005}, pages={1–64} }
 @article{baiti_bressan_jenssen_2005, title={Instability of travelling wave profiles for the Lax-Friedrichs scheme}, volume={13}, number={4}, journal={Discrete and Continuous Dynamical Systems}, author={Baiti, P. and Bressan, A. and Jenssen, H. K.}, year={2005}, pages={877–899} }
 @article{jenssen_young_2004, title={Gradient driven and singular flux blowup of smooth solutions to hyperbolic systems of conservation laws}, volume={1}, ISSN={["1793-6993"]}, DOI={10.1142/S021989160400024X}, abstractNote={We consider two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws. The explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function. The examples show that solutions of uniformly strictly hyperbolic systems can remain as smooth as the initial data until the time of blowup. Consequently, blowup in amplitude is not necessarily strictly preceded by shock formation.}, number={4}, journal={JOURNAL OF HYPERBOLIC DIFFERENTIAL EQUATIONS}, author={Jenssen, HK and Young, R}, year={2004}, month={Dec}, pages={627–641} }
 @article{hoff_jenssen_2004, title={Symmetric nonbarotropic flows with large data and forces}, volume={173}, ISSN={["0003-9527"]}, DOI={10.1007/s00205-004-0318-5}, abstractNote={We prove the global existence of weak solutions of the Navier-Stokes equations of compressible, nonbarotropic flow in three space dimensions with initial data and external forces which are large, discontinuous, and spherically or cylindrically symmetric. The analysis allows for the possibility that a vacuum state emerges at the origin or axis of symmetry, and the equations hold in the sense of distributions in the set where the density is positive. In addition, the mass and momentum equations hold weakly in the entire space-time domain, but with a nonstandard interpretation of the viscosity terms as distributions. Solutions are obtained as limits of solutions in annular regions between two balls or cylinders, and the analysis allows for the possibility that energy is absorbed into the origin or axis, and is lost in the limit as the inner radius goes to zero.}, number={3}, journal={ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS}, author={Hoff, D and Jenssen, HK}, year={2004}, month={Sep}, pages={297–343} }